Algebraic system, automorphism of an

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 $C_7$" Quart. J. Math. Oxford (2) , 18 (1967) pp. 53–57