Namespaces
Variants
Actions

Difference between revisions of "Algebraic system, automorphism of an"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
An isomorphic mapping of an algebraic system onto itself. An automorphism of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a0116602.png" />-system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a0116603.png" /> is a one-to-one mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a0116604.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a0116605.png" /> onto itself having the following properties:
+
<!--
 +
a0116602.png
 +
$#A+1 = 141 n = 1
 +
$#C+1 = 141 : ~/encyclopedia/old_files/data/A011/A.0101660 Algebraic system, automorphism of an
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a0116606.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a0116607.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
An isomorphic mapping of an algebraic system onto itself. An automorphism of an  $  \Omega $-
 +
system  $  \mathbf A = \langle  A, \Omega \rangle $
 +
is a one-to-one mapping  $  \phi $
 +
of the set  $  A $
 +
onto itself having the following properties:
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a0116608.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a0116609.png" /> and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166010.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166011.png" />. In other words, an automorphism of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166012.png" />-system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166013.png" /> is an isomorphic mapping of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166014.png" /> onto itself. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166015.png" /> be the set of all automorphisms of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166016.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166017.png" />, the inverse mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166018.png" /> also has the properties (1) and (2), and for this reason <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166019.png" />. The product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166020.png" /> of two automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166021.png" /> of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166022.png" />, defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166024.png" />, is again an automorphism of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166025.png" />. Since multiplication of mappings is associative, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166026.png" /> is a group, known as the group of all automorphisms of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166028.png" />; it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166029.png" />. The subgroups of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166030.png" /> are simply called automorphism groups of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166032.png" />.
+
$$ \tag{1 }
 +
\phi ( F ( x _ {1} \dots x _ {n} ) )  = F ( \phi ( x _ {1} ) \dots
 +
\phi ( x _ {n} ) ) ,
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166033.png" /> be an automorphism of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166034.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166035.png" /> be a congruence of this system. Putting
+
$$ \tag{2 }
 +
P ( x _ {1} \dots x _ {m} )  \iff  P ( \phi ( x _ {1} ) \dots \phi ( x _ {m} ) ),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166036.png" /></td> </tr></table>
+
for all  $  x _ {1} , x _ {2} \dots $
 +
from  $  A $
 +
and for all  $  F, P $
 +
from  $  \Omega $.
 +
In other words, an automorphism of an  $  \Omega $-
 +
system  $  \mathbf A $
 +
is an isomorphic mapping of the system  $  \mathbf A $
 +
onto itself. Let  $  G $
 +
be the set of all automorphisms of the system  $  \mathbf A $.
 +
If  $  \phi \in G $,
 +
the inverse mapping  $  \phi  ^ {-1} $
 +
also has the properties (1) and (2), and for this reason  $  \phi  ^ {-1} \in G $.
 +
The product  $  \alpha = \phi \psi $
 +
of two automorphisms  $  \phi , \psi $
 +
of the system  $  \mathbf A $,
 +
defined by the formula  $  \alpha (x) = \psi ( \phi (x) ) $,
 +
$  x \in A $,
 +
is again an automorphism of the system  $  \mathbf A $.
 +
Since multiplication of mappings is associative,  $  \langle  G, \cdot , {}  ^ {-1} \rangle $
 +
is a group, known as the group of all automorphisms of the system  $  \mathbf A $;  
 +
it is denoted by  $  \mathop{\rm Aut} ( \mathbf A ) $.  
 +
The subgroups of the group  $  \mathop{\rm Aut} ( \mathbf A ) $
 +
are simply called automorphism groups of the system  $  \mathbf A $.
  
one again obtains a congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166037.png" /> of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166038.png" />. The automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166039.png" /> is known as an IC-automorphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166040.png" /> for any congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166041.png" /> of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166042.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166043.png" /> of all IC-automorphisms of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166044.png" /> is a normal subgroup of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166045.png" />, and the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166046.png" /> is isomorphic to an automorphism group of the lattice of all congruences of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166047.png" /> [[#References|[1]]]. In particular, any inner automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166048.png" /> of a group defined by a fixed element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166049.png" /> of this group is an IC-automorphism. However, the example of a cyclic group of prime order shows that not all IC-automorphisms of a group are inner.
+
Let  $  \phi $
 +
be an automorphism of the system $  \mathbf A $
 +
and let  $  \theta $
 +
be a congruence of this system. Putting
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166050.png" /> be a non-trivial variety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166051.png" />-systems or any other class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166052.png" />-systems comprising free systems of any (non-zero) rank. An automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166053.png" /> of a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166054.png" /> of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166055.png" /> is called an I-automorphism if there exists a term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166056.png" /> of the signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166057.png" />, in the unknowns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166058.png" />, for which: 1) in the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166059.png" /> there exist elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166060.png" /> such that for each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166061.png" /> the equality
+
$$
 +
( x , y ) \in \theta _  \phi  \iff  ( \phi  ^ {-1} ( x ) ,\
 +
\phi  ^ {-1} ( y ) ) \in \theta ,\  x , y \in \mathbf A ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166062.png" /></td> </tr></table>
+
one again obtains a congruence  $  \theta _  \phi  $
 +
of the system  $  \mathbf A $.
 +
The automorphism  $  \phi $
 +
is known as an IC-automorphism if  $  \theta _  \phi  = \theta $
 +
for any congruence  $  \theta $
 +
of the system  $  \mathbf A $.
 +
The set  $  \mathop{\rm IC} ( \mathbf A ) $
 +
of all IC-automorphisms of the system  $  \mathbf A $
 +
is a normal subgroup of the group  $  \mathop{\rm Aut} ( \mathbf A ) $,
 +
and the quotient group  $  \mathop{\rm Aut} ( \mathbf A ) / \mathop{\rm IC} ( \mathbf A ) $
 +
is isomorphic to an automorphism group of the lattice of all congruences of the system  $  \mathbf A $[[#References|[1]]]. In particular, any inner automorphism  $  x \rightarrow a  ^ {-1} xa $
 +
of a group defined by a fixed element  $  a $
 +
of this group is an IC-automorphism. However, the example of a cyclic group of prime order shows that not all IC-automorphisms of a group are inner.
  
is valid; and 2) for any system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166063.png" /> of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166064.png" /> the mapping
+
Let  $  \mathfrak K $
 +
be a non-trivial variety of  $  \Omega $-
 +
systems or any other class of  $  \Omega $-
 +
systems comprising free systems of any (non-zero) rank. An automorphism  $  \phi $
 +
of a system  $  \mathbf A $
 +
of the class $  \mathfrak K $
 +
is called an I-automorphism if there exists a term  $  f _  \phi  (x _ {1} \dots x _ {n} ) $
 +
of the signature  $  \Omega $,
 +
in the unknowns  $  x _ {1} \dots x _ {n} $,
 +
for which: 1) in the system  $  \mathbf A $
 +
there exist elements  $  a _ {2} \dots a _ {n} $
 +
such that for each element  $  x \in A $
 +
the equality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166065.png" /></td> </tr></table>
+
$$
 +
\phi ( x )  = f _  \phi  ( x , a _ {2} \dots a _ {n} )
 +
$$
  
is an automorphism of this system for any arbitrary selection of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166066.png" /> in the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166067.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166068.png" /> of all I-automorphisms for each system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166069.png" /> of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166070.png" /> is a normal subgroup of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166071.png" />. In the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166072.png" /> of all groups the concept of an I-automorphism coincides with the concept of an inner automorphism of the group [[#References|[2]]]. For the more general concept of a formula automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166074.png" />-systems, see [[#References|[3]]].
+
is valid; and 2) for any system $  \mathbf B $
 +
of the class $  \mathfrak K $
 +
the mapping
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166075.png" /> be an algebraic system. By replacing each basic operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166076.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166077.png" /> by the predicate
+
$$
 +
x  \rightarrow  f _  \phi  ( x , x _ {2} \dots x _ {n} )
 +
\  ( x \in B )
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166078.png" /></td> </tr></table>
+
is an automorphism of this system for any arbitrary selection of elements  $  x _ {2} \dots x _ {n} $
 +
in the system  $  \mathbf B $.
 +
The set  $  \textrm{ I } ( \mathbf A ) $
 +
of all I-automorphisms for each system  $  \mathbf A $
 +
of the class $  \mathfrak K $
 +
is a normal subgroup of the group  $  \mathop{\rm Aut} ( \mathbf A ) $.
 +
In the class  $  \mathfrak K $
 +
of all groups the concept of an I-automorphism coincides with the concept of an inner automorphism of the group [[#References|[2]]]. For the more general concept of a formula automorphism of  $  \Omega $-
 +
systems, see [[#References|[3]]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166079.png" /></td> </tr></table>
+
Let  $  \mathbf A $
 +
be an algebraic system. By replacing each basic operation  $  F $
 +
in  $  \mathbf A $
 +
by the predicate
  
one obtains the so-called model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166081.png" /> which represents the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166082.png" />. The equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166083.png" /> is valid. If the systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166085.png" /> have a common carrier <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166086.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166087.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166088.png" />. If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166089.png" />-system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166090.png" /> with a finite number of generators is finitely approximable, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166091.png" /> is also finitely approximable (cf. [[#References|[1]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166092.png" /> be a class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166093.png" />-systems and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166094.png" /> be the class of all isomorphic copies of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166096.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166097.png" /> be the class of subgroups of groups from the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166098.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a01166099.png" /> consists of groups which are isomorphically imbeddable into the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660101.png" />.
+
$$
 +
R ( x _ {1} \dots x _ {n} , y )  \iff \
 +
F ( x _ {1} \dots x _ {n} ) = y
 +
$$
 +
 
 +
$$
 +
( x _ {1} \dots x _ {n} , y \in A ) ,
 +
$$
 +
 
 +
one obtains the so-called model $  \mathbf A  ^ {*} $
 +
which represents the system $  \mathbf A $.  
 +
The equality $  \mathop{\rm Aut} ( \mathbf A  ^ {*} ) = \mathop{\rm Aut} ( \mathbf A ) $
 +
is valid. If the systems $  \mathbf A = \langle  A, \Omega \rangle $
 +
and $  \mathbf A  ^  \prime  = \langle  A , \Omega  ^  \prime  \rangle $
 +
have a common carrier $  A $,  
 +
and if $  \Omega \subset  \Omega  ^  \prime  $,  
 +
then $  \mathop{\rm Aut} ( \mathbf A ) \supseteq  \mathop{\rm Aut} ( \mathbf A  ^  \prime  ) $.  
 +
If the $  \Omega $-
 +
system $  \mathbf A $
 +
with a finite number of generators is finitely approximable, the group $  \mathop{\rm Aut} ( \mathbf A ) $
 +
is also finitely approximable (cf. [[#References|[1]]]). Let $  \mathfrak K $
 +
be a class of $  \Omega $-
 +
systems and let $  \mathop{\rm Aut} ( \mathfrak K ) $
 +
be the class of all isomorphic copies of the groups $  \mathop{\rm Aut} ( \mathbf A ) $,  
 +
$  \mathbf A \in \mathfrak K $,  
 +
and let $  \mathop{\rm SAut} ( \mathfrak K ) $
 +
be the class of subgroups of groups from the class $  \mathop{\rm Aut} ( \mathfrak K ) $.  
 +
The class $  \mathop{\rm SAut} ( \mathfrak K ) $
 +
consists of groups which are isomorphically imbeddable into the groups $  \mathop{\rm Aut} ( \mathbf A ) $,  
 +
$  \mathbf A \in \mathfrak K $.
  
 
The following two problems arose in the study of automorphism groups of algebraic systems.
 
The following two problems arose in the study of automorphism groups of algebraic systems.
  
1) Given a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660102.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660103.png" />-systems, what can one say about the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660105.png" />?
+
1) Given a class $  \mathfrak K $
 +
of $  \Omega $-
 +
systems, what can one say about the classes $  \mathop{\rm Aut} ( \mathfrak K ) $
 +
and $  \mathop{\rm SAut} ( \mathfrak K ) $?
  
2) Let an (abstract) class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660106.png" /> of groups be given. Does there exist a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660107.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660108.png" />-systems with a given signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660109.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660110.png" /> or even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660111.png" />? It has been proved that for any axiomatizable class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660112.png" /> of models the class of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660113.png" /> is universally axiomatizable [[#References|[1]]]. It has also been proved [[#References|[1]]], [[#References|[4]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660114.png" /> is an axiomatizable class of models comprising infinite models, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660115.png" /> is a totally ordered set and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660116.png" /> is an automorphism group of the model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660117.png" />, then there exists a model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660118.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660119.png" />, and for each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660120.png" /> there exists an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660121.png" /> of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660122.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660123.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660124.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660125.png" /> is called 1) universal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660126.png" /> for any axiomatizable class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660127.png" /> of models comprising infinite models; and 2) a group of ordered automorphisms of an ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660129.png" /> (cf. [[Totally ordered group|Totally ordered group]]) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660130.png" /> is isomorphic to some automorphism group of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660131.png" /> which preserves the given total order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660132.png" /> of this group (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660133.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660134.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660135.png" />).
+
2) Let an (abstract) class $  K $
 +
of groups be given. Does there exist a class $  \mathfrak K $
 +
of $  \Omega $-
 +
systems with a given signature $  \Omega $
 +
such that $  K = \mathop{\rm Aut} ( \mathfrak K ) $
 +
or even $  K = \mathop{\rm SAut} ( \mathfrak K ) $?  
 +
It has been proved that for any axiomatizable class $  \mathfrak K $
 +
of models the class of groups $  \mathop{\rm SAut} ( \mathfrak K ) $
 +
is universally axiomatizable [[#References|[1]]]. It has also been proved [[#References|[1]]], [[#References|[4]]] that if $  \mathfrak K $
 +
is an axiomatizable class of models comprising infinite models, if $  \langle  B, \leq  \rangle $
 +
is a totally ordered set and if $  \mathbf G $
 +
is an automorphism group of the model $  \langle  B, \leq  \rangle $,  
 +
then there exists a model $  \mathbf A \in \mathfrak K $
 +
such that $  A \supseteq B $,  
 +
and for each element $  g \in G $
 +
there exists an automorphism $  \phi $
 +
of the system $  \mathbf A $
 +
such that $  g(x) = \phi (x) $
 +
for all $  x \in B $.  
 +
The group $  G $
 +
is called 1) universal if $  G \in  \mathop{\rm SAut} ( \mathfrak K ) $
 +
for any axiomatizable class $  \mathfrak K $
 +
of models comprising infinite models; and 2) a group of ordered automorphisms of an ordered group $  \mathbf H $(
 +
cf. [[Totally ordered group|Totally ordered group]]) if $  \mathbf G $
 +
is isomorphic to some automorphism group of the group $  \mathbf H $
 +
which preserves the given total order $  \leq  $
 +
of this group (i.e. $  a \leq  b \Rightarrow \phi (a) \leq  \phi (b) $
 +
for all a, b \in H $,
 +
$  \phi \in G $).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660136.png" /> be the class of totally ordered sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660137.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660138.png" /> be the class of universal groups, let RO be the class of right-ordered groups and let OA be the class of ordered automorphism groups of free Abelian groups. Then [[#References|[4]]], [[#References|[5]]], [[#References|[6]]]:
+
Let $  l $
 +
be the class of totally ordered sets $  \langle  M, \leq  \rangle $,
 +
let $  \mathfrak U $
 +
be the class of universal groups, let RO be the class of right-ordered groups and let OA be the class of ordered automorphism groups of free Abelian groups. Then [[#References|[4]]], [[#References|[5]]], [[#References|[6]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660139.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm SAut} ( l )  = \mathfrak U  =   \mathop{\rm RO}  =   \mathop{\rm OA} .
 +
$$
  
Each group is isomorphic to the group of all automorphisms of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660140.png" />-algebra. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660141.png" /> is the class of all rings, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660142.png" /> is the class of all groups [[#References|[1]]]. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660143.png" /> is the class of all groups, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660144.png" />; for example, the cyclic groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660145.png" /> of the respective orders 3, 5 and 7 do not belong to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660146.png" />. There is also no topological group whose group of all topological automorphisms is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660147.png" /> [[#References|[7]]].
+
Each group is isomorphic to the group of all automorphisms of some $  \Omega $-
 +
algebra. If $  \mathfrak K $
 +
is the class of all rings, $  \mathop{\rm Aut} ( \mathfrak K ) $
 +
is the class of all groups [[#References|[1]]]. However, if $  \mathfrak K $
 +
is the class of all groups, $  \mathop{\rm Aut} ( \mathfrak K ) \neq \mathfrak K $;  
 +
for example, the cyclic groups $  \mathbf C _ {3} , \mathbf C _ {5} , \mathbf C _ {7} $
 +
of the respective orders 3, 5 and 7 do not belong to the class $  \mathop{\rm Aut} ( \mathfrak K ) $.  
 +
There is also no topological group whose group of all topological automorphisms is isomorphic to $  \mathbf C _ {5} $[[#References|[7]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.I. Plotkin,  "Groups of automorphisms of algebraic systems" , Wolters-Noordhoff  (1972)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B. Csákány,  "Inner automorphisms of universal algebras"  ''Publ. Math. Debrecen'' , '''12'''  (1965)  pp. 331–333</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Grant,  "Automorphisms definable by formulas"  ''Pacific J. Math.'' , '''44'''  (1973)  pp. 107–115</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.O. Rabin,  "Universal groups of automorphisms of models" , ''Theory of models'' , North-Holland  (1965)  pp. 274–284</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.M. Cohn,  "Groups of order automorphisms of ordered sets"  ''Mathematika'' , '''4'''  (1957)  pp. 41–50</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D.M. Smirnov,  "Right-ordered groups"  ''Algebra i Logika'' , '''5''' :  6  (1966)  pp. 41–59  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  R.J. Wille,  "The existence of a topological group with automorphism group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660148.png" />"  ''Quart. J. Math. Oxford (2)'' , '''18'''  (1967)  pp. 53–57</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.I. Plotkin,  "Groups of automorphisms of algebraic systems" , Wolters-Noordhoff  (1972)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B. Csákány,  "Inner automorphisms of universal algebras"  ''Publ. Math. Debrecen'' , '''12'''  (1965)  pp. 331–333</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Grant,  "Automorphisms definable by formulas"  ''Pacific J. Math.'' , '''44'''  (1973)  pp. 107–115</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.O. Rabin,  "Universal groups of automorphisms of models" , ''Theory of models'' , North-Holland  (1965)  pp. 274–284</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.M. Cohn,  "Groups of order automorphisms of ordered sets"  ''Mathematika'' , '''4'''  (1957)  pp. 41–50</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D.M. Smirnov,  "Right-ordered groups"  ''Algebra i Logika'' , '''5''' :  6  (1966)  pp. 41–59  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  R.J. Wille,  "The existence of a topological group with automorphism group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011660/a011660148.png" />"  ''Quart. J. Math. Oxford (2)'' , '''18'''  (1967)  pp. 53–57</TD></TR></table>

Latest revision as of 16:10, 1 April 2020


An isomorphic mapping of an algebraic system onto itself. An automorphism of an $ \Omega $- system $ \mathbf A = \langle A, \Omega \rangle $ is a one-to-one mapping $ \phi $ of the set $ A $ onto itself having the following properties:

$$ \tag{1 } \phi ( F ( x _ {1} \dots x _ {n} ) ) = F ( \phi ( x _ {1} ) \dots \phi ( x _ {n} ) ) , $$

$$ \tag{2 } P ( x _ {1} \dots x _ {m} ) \iff P ( \phi ( x _ {1} ) \dots \phi ( x _ {m} ) ), $$

for all $ x _ {1} , x _ {2} \dots $ from $ A $ and for all $ F, P $ from $ \Omega $. In other words, an automorphism of an $ \Omega $- system $ \mathbf A $ is an isomorphic mapping of the system $ \mathbf A $ onto itself. Let $ G $ be the set of all automorphisms of the system $ \mathbf A $. If $ \phi \in G $, the inverse mapping $ \phi ^ {-1} $ also has the properties (1) and (2), and for this reason $ \phi ^ {-1} \in G $. The product $ \alpha = \phi \psi $ of two automorphisms $ \phi , \psi $ of the system $ \mathbf A $, defined by the formula $ \alpha (x) = \psi ( \phi (x) ) $, $ x \in A $, is again an automorphism of the system $ \mathbf A $. Since multiplication of mappings is associative, $ \langle G, \cdot , {} ^ {-1} \rangle $ is a group, known as the group of all automorphisms of the system $ \mathbf A $; it is denoted by $ \mathop{\rm Aut} ( \mathbf A ) $. The subgroups of the group $ \mathop{\rm Aut} ( \mathbf A ) $ are simply called automorphism groups of the system $ \mathbf A $.

Let $ \phi $ be an automorphism of the system $ \mathbf A $ and let $ \theta $ be a congruence of this system. Putting

$$ ( x , y ) \in \theta _ \phi \iff ( \phi ^ {-1} ( x ) ,\ \phi ^ {-1} ( y ) ) \in \theta ,\ x , y \in \mathbf A , $$

one again obtains a congruence $ \theta _ \phi $ of the system $ \mathbf A $. The automorphism $ \phi $ is known as an IC-automorphism if $ \theta _ \phi = \theta $ for any congruence $ \theta $ of the system $ \mathbf A $. The set $ \mathop{\rm IC} ( \mathbf A ) $ of all IC-automorphisms of the system $ \mathbf A $ is a normal subgroup of the group $ \mathop{\rm Aut} ( \mathbf A ) $, and the quotient group $ \mathop{\rm Aut} ( \mathbf A ) / \mathop{\rm IC} ( \mathbf A ) $ is isomorphic to an automorphism group of the lattice of all congruences of the system $ \mathbf A $[1]. In particular, any inner automorphism $ x \rightarrow a ^ {-1} xa $ of a group defined by a fixed element $ a $ of this group is an IC-automorphism. However, the example of a cyclic group of prime order shows that not all IC-automorphisms of a group are inner.

Let $ \mathfrak K $ be a non-trivial variety of $ \Omega $- systems or any other class of $ \Omega $- systems comprising free systems of any (non-zero) rank. An automorphism $ \phi $ of a system $ \mathbf A $ of the class $ \mathfrak K $ is called an I-automorphism if there exists a term $ f _ \phi (x _ {1} \dots x _ {n} ) $ of the signature $ \Omega $, in the unknowns $ x _ {1} \dots x _ {n} $, for which: 1) in the system $ \mathbf A $ there exist elements $ a _ {2} \dots a _ {n} $ such that for each element $ x \in A $ the equality

$$ \phi ( x ) = f _ \phi ( x , a _ {2} \dots a _ {n} ) $$

is valid; and 2) for any system $ \mathbf B $ of the class $ \mathfrak K $ the mapping

$$ x \rightarrow f _ \phi ( x , x _ {2} \dots x _ {n} ) \ ( x \in B ) $$

is an automorphism of this system for any arbitrary selection of elements $ x _ {2} \dots x _ {n} $ in the system $ \mathbf B $. The set $ \textrm{ I } ( \mathbf A ) $ of all I-automorphisms for each system $ \mathbf A $ of the class $ \mathfrak K $ is a normal subgroup of the group $ \mathop{\rm Aut} ( \mathbf A ) $. In the class $ \mathfrak K $ of all groups the concept of an I-automorphism coincides with the concept of an inner automorphism of the group [2]. For the more general concept of a formula automorphism of $ \Omega $- systems, see [3].

Let $ \mathbf A $ be an algebraic system. By replacing each basic operation $ F $ in $ \mathbf A $ by the predicate

$$ R ( x _ {1} \dots x _ {n} , y ) \iff \ F ( x _ {1} \dots x _ {n} ) = y $$

$$ ( x _ {1} \dots x _ {n} , y \in A ) , $$

one obtains the so-called model $ \mathbf A ^ {*} $ which represents the system $ \mathbf A $. The equality $ \mathop{\rm Aut} ( \mathbf A ^ {*} ) = \mathop{\rm Aut} ( \mathbf A ) $ is valid. If the systems $ \mathbf A = \langle A, \Omega \rangle $ and $ \mathbf A ^ \prime = \langle A , \Omega ^ \prime \rangle $ have a common carrier $ A $, and if $ \Omega \subset \Omega ^ \prime $, then $ \mathop{\rm Aut} ( \mathbf A ) \supseteq \mathop{\rm Aut} ( \mathbf A ^ \prime ) $. If the $ \Omega $- system $ \mathbf A $ with a finite number of generators is finitely approximable, the group $ \mathop{\rm Aut} ( \mathbf A ) $ is also finitely approximable (cf. [1]). Let $ \mathfrak K $ be a class of $ \Omega $- systems and let $ \mathop{\rm Aut} ( \mathfrak K ) $ be the class of all isomorphic copies of the groups $ \mathop{\rm Aut} ( \mathbf A ) $, $ \mathbf A \in \mathfrak K $, and let $ \mathop{\rm SAut} ( \mathfrak K ) $ be the class of subgroups of groups from the class $ \mathop{\rm Aut} ( \mathfrak K ) $. The class $ \mathop{\rm SAut} ( \mathfrak K ) $ consists of groups which are isomorphically imbeddable into the groups $ \mathop{\rm Aut} ( \mathbf A ) $, $ \mathbf A \in \mathfrak K $.

The following two problems arose in the study of automorphism groups of algebraic systems.

1) Given a class $ \mathfrak K $ of $ \Omega $- systems, what can one say about the classes $ \mathop{\rm Aut} ( \mathfrak K ) $ and $ \mathop{\rm SAut} ( \mathfrak K ) $?

2) Let an (abstract) class $ K $ of groups be given. Does there exist a class $ \mathfrak K $ of $ \Omega $- systems with a given signature $ \Omega $ such that $ K = \mathop{\rm Aut} ( \mathfrak K ) $ or even $ K = \mathop{\rm SAut} ( \mathfrak K ) $? It has been proved that for any axiomatizable class $ \mathfrak K $ of models the class of groups $ \mathop{\rm SAut} ( \mathfrak K ) $ is universally axiomatizable [1]. It has also been proved [1], [4] that if $ \mathfrak K $ is an axiomatizable class of models comprising infinite models, if $ \langle B, \leq \rangle $ is a totally ordered set and if $ \mathbf G $ is an automorphism group of the model $ \langle B, \leq \rangle $, then there exists a model $ \mathbf A \in \mathfrak K $ such that $ A \supseteq B $, and for each element $ g \in G $ there exists an automorphism $ \phi $ of the system $ \mathbf A $ such that $ g(x) = \phi (x) $ for all $ x \in B $. The group $ G $ is called 1) universal if $ G \in \mathop{\rm SAut} ( \mathfrak K ) $ for any axiomatizable class $ \mathfrak K $ of models comprising infinite models; and 2) a group of ordered automorphisms of an ordered group $ \mathbf H $( cf. Totally ordered group) if $ \mathbf G $ is isomorphic to some automorphism group of the group $ \mathbf H $ which preserves the given total order $ \leq $ of this group (i.e. $ a \leq b \Rightarrow \phi (a) \leq \phi (b) $ for all $ a, b \in H $, $ \phi \in G $).

Let $ l $ be the class of totally ordered sets $ \langle M, \leq \rangle $, let $ \mathfrak U $ be the class of universal groups, let RO be the class of right-ordered groups and let OA be the class of ordered automorphism groups of free Abelian groups. Then [4], [5], [6]:

$$ \mathop{\rm SAut} ( l ) = \mathfrak U = \mathop{\rm RO} = \mathop{\rm OA} . $$

Each group is isomorphic to the group of all automorphisms of some $ \Omega $- algebra. If $ \mathfrak K $ is the class of all rings, $ \mathop{\rm Aut} ( \mathfrak K ) $ is the class of all groups [1]. However, if $ \mathfrak K $ is the class of all groups, $ \mathop{\rm Aut} ( \mathfrak K ) \neq \mathfrak K $; for example, the cyclic groups $ \mathbf C _ {3} , \mathbf C _ {5} , \mathbf C _ {7} $ of the respective orders 3, 5 and 7 do not belong to the class $ \mathop{\rm Aut} ( \mathfrak K ) $. There is also no topological group whose group of all topological automorphisms is isomorphic to $ \mathbf C _ {5} $[7].

References

[1] B.I. Plotkin, "Groups of automorphisms of algebraic systems" , Wolters-Noordhoff (1972)
[2] B. Csákány, "Inner automorphisms of universal algebras" Publ. Math. Debrecen , 12 (1965) pp. 331–333
[3] J. Grant, "Automorphisms definable by formulas" Pacific J. Math. , 44 (1973) pp. 107–115
[4] M.O. Rabin, "Universal groups of automorphisms of models" , Theory of models , North-Holland (1965) pp. 274–284
[5] P.M. Cohn, "Groups of order automorphisms of ordered sets" Mathematika , 4 (1957) pp. 41–50
[6] D.M. Smirnov, "Right-ordered groups" Algebra i Logika , 5 : 6 (1966) pp. 41–59 (In Russian)
[7] R.J. Wille, "The existence of a topological group with automorphism group " Quart. J. Math. Oxford (2) , 18 (1967) pp. 53–57
How to Cite This Entry:
Algebraic system, automorphism of an. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_system,_automorphism_of_an&oldid=45065
This article was adapted from an original article by D.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article