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A class of all groups satisfying a fixed system of identity relations, or laws,
 
A class of all groups satisfying a fixed system of identity relations, or laws,
  
<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/v/v096/v096290/v0962901.png" /></td> </tr></table>
+
$$
 +
v ( x _ {1} \dots x _ {n} )  = 1,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v0962902.png" /> runs through some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v0962903.png" /> of group words, i.e. elements of the [[Free group|free group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v0962904.png" /> with free generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v0962905.png" />. Just like any variety of algebraic systems (cf. [[Algebraic systems, variety of|Algebraic systems, variety of]]), a variety of groups can also be defined by the property of being closed under subsystems (subgroups), homomorphic images and Cartesian products. The smallest variety containing a given class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v0962906.png" /> of groups is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v0962907.png" />. Regarding the operations of intersection and union of varieties, defined by the formula
+
where v $
 +
runs through some set $  V $
 +
of group words, i.e. elements of the [[Free group|free group]] $  X $
 +
with free generators $  x _ {1} \dots x _ {n} , . . . $.  
 +
Just like any variety of algebraic systems (cf. [[Algebraic systems, variety of|Algebraic systems, variety of]]), a variety of groups can also be defined by the property of being closed under subsystems (subgroups), homomorphic images and Cartesian products. The smallest variety containing a given class $  \mathfrak C $
 +
of groups is denoted by $  \mathop{\rm var}  \mathfrak C $.  
 +
Regarding the operations of intersection and union of varieties, defined by the formula
  
<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/v/v096/v096290/v0962908.png" /></td> </tr></table>
+
$$
 +
\mathfrak U \lor \mathfrak V  =   \mathop{\rm var}  ( \mathfrak U \cup \mathfrak V ),
 +
$$
  
varieties of groups form a complete modular, but not distributive, lattice. The product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v0962909.png" /> of two varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629011.png" /> is defined as the variety of groups consisting of all groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629012.png" /> with a normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629014.png" />. Any variety of groups other than the variety of trivial groups and the variety of all groups can be uniquely represented as a product of varieties of groups which cannot be split further.
+
varieties of groups form a complete modular, but not distributive, lattice. The product $  \mathfrak U \mathfrak V $
 +
of two varieties $  \mathfrak U $
 +
and $  \mathfrak V $
 +
is defined as the variety of groups consisting of all groups $  G $
 +
with a normal subgroup $  N \in \mathfrak U $
 +
such that $  G/N \in \mathfrak V $.  
 +
Any variety of groups other than the variety of trivial groups and the variety of all groups can be uniquely represented as a product of varieties of groups which cannot be split further.
  
Examples of varieties of groups: the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629015.png" /> of all Abelian groups; the Burnside variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629016.png" /> of all groups of exponent (index) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629017.png" />, defined by the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629018.png" />; the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629019.png" />; the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629020.png" /> of all nilpotent groups of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629021.png" />; the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629022.png" /> of all solvable groups of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629023.png" />; in particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629025.png" /> is the variety of metabelian groups.
+
Examples of varieties of groups: the variety $  \mathfrak A $
 +
of all Abelian groups; the Burnside variety $  \mathfrak B _ {n} $
 +
of all groups of exponent (index) $  n $,  
 +
defined by the identity $  x  ^ {n} = 1 $;  
 +
the variety $  \mathfrak A _ {n} = \mathfrak B _ {n} \wedge \mathfrak A $;  
 +
the variety $  \mathfrak N _ {c} $
 +
of all nilpotent groups of class $  \leq  c $;  
 +
the variety $  \mathfrak A  ^ {l} $
 +
of all solvable groups of length $  \leq  l $;  
 +
in particular, if $  l = 2 $,  
 +
$  \mathfrak A  ^ {2} $
 +
is the variety of metabelian groups.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629026.png" /> be some property of groups. One says that a variety of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629027.png" /> has the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629028.png" /> (locally) if each (finitely-generated) group in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629029.png" /> has the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629030.png" />. One says, in this exact sense, that the variety is nilpotent, locally nilpotent, locally finite, etc.
+
Let $  {\mathcal P} $
 +
be some property of groups. One says that a variety of groups $  \mathfrak V $
 +
has the property $  {\mathcal P} $(
 +
locally) if each (finitely-generated) group in $  \mathfrak V $
 +
has the property $  {\mathcal P} $.  
 +
One says, in this exact sense, that the variety is nilpotent, locally nilpotent, locally finite, etc.
  
The properties of a solvable variety of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629031.png" /> depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629032.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629033.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629034.png" /> for certain suitable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629036.png" /> [[#References|[2]]], [[#References|[3]]]. The description of metabelian varieties of groups is reduced, to a large extent, to the description of locally finite varieties of groups: If a metabelian variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629037.png" /> is not locally finite, then
+
The properties of a solvable variety of groups $  \mathfrak V $
 +
depend on $  \mathfrak V \wedge \mathfrak A  ^ {2} $.  
 +
Thus, if $  \mathfrak B \supseteq \mathfrak A  ^ {2} $,  
 +
then $  \mathfrak V \subseteq \mathfrak B _ {n} \mathfrak N _ {c} \mathfrak B _ {n} $
 +
for certain suitable $  n $
 +
and $  c $[[#References|[2]]], [[#References|[3]]]. The description of metabelian varieties of groups is reduced, to a large extent, to the description of locally finite varieties of groups: If a metabelian variety $  \mathfrak V $
 +
is not locally finite, then
  
<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/v/v096/v096290/v09629038.png" /></td> </tr></table>
+
$$
 +
\mathfrak B  = \
 +
\mathfrak B _ {1} \lor \mathfrak B _ {2} \lor \mathfrak B _ {3} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629040.png" /> is uniquely representable as the union of a finite number of varieties of groups of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629041.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629042.png" /> is locally finite [[#References|[4]]]. Certain locally finite metabelian varieties have been described — for example, varieties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629043.png" />-groups of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629044.png" /> (cf. [[#References|[5]]]).
+
where $  \mathfrak B _ {1} = \mathfrak A _ {m} \mathfrak A $,  
 +
$  \mathfrak V _ {2} $
 +
is uniquely representable as the union of a finite number of varieties of groups of the form $  \mathfrak N _ {c} \mathfrak A _ {k} \wedge \mathfrak A  ^ {2} $,  
 +
and $  \mathfrak V _ {3} $
 +
is locally finite [[#References|[4]]]. Certain locally finite metabelian varieties have been described — for example, varieties of $  p $-
 +
groups of class $  \leq  p + 1 $(
 +
cf. [[#References|[5]]]).
  
A variety of groups is said to be a Cross variety if it is generated by a finite group. Cross varieties of groups are locally finite. A variety of groups is said to be a near Cross variety if it is not Cross, but each of its proper subvarieties is Cross. The solvable near Cross varieties are exhausted by the varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629049.png" /> are different prime numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629050.png" /> for odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629052.png" /> [[#References|[6]]]. There exist, however, other near Cross varieties; such varieties are contained, for example, in any variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629053.png" /> of all locally finite groups of exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629054.png" /> [[#References|[7]]]. An important role in the study of locally finite varieties of groups is played by critical groups — groups not contained in the variety generated by all their proper subgroups and quotient groups. A Cross variety can contain only a finite number of non-isomorphic critical groups. All locally finite varieties are generated by their critical groups.
+
A variety of groups is said to be a Cross variety if it is generated by a finite group. Cross varieties of groups are locally finite. A variety of groups is said to be a near Cross variety if it is not Cross, but each of its proper subvarieties is Cross. The solvable near Cross varieties are exhausted by the varieties $  \mathfrak A $,  
 +
$  \mathfrak A _ {p}  ^ {2} $,  
 +
$  \mathfrak A _ {p} \mathfrak A _ {q} \mathfrak A _ {r} $,  
 +
$  \mathfrak A _ {p} \mathfrak T _ {q} $,  
 +
where $  p, q, r $
 +
are different prime numbers, $  \mathfrak T _ {q} = \mathfrak B _ {q} \wedge \mathfrak N _ {2} $
 +
for odd $  q $
 +
and $  \mathfrak T _ {2} = \mathfrak B _ {4} \wedge \mathfrak N _ {2} $[[#References|[6]]]. There exist, however, other near Cross varieties; such varieties are contained, for example, in any variety $  \mathfrak K $
 +
of all locally finite groups of exponent $  p \geq  5 $[[#References|[7]]]. An important role in the study of locally finite varieties of groups is played by critical groups — groups not contained in the variety generated by all their proper subgroups and quotient groups. A Cross variety can contain only a finite number of non-isomorphic critical groups. All locally finite varieties are generated by their critical groups.
  
A variety of groups is said to be finitely based if it can be specified by a given finite number of identities. These include, for example, all Cross, nilpotent and metabelian varieties. It has been proved [[#References|[8]]] that non-finitely based varieties of groups exist, and that the number of all varieties of groups has the power of the continuum. For examples of infinite independent systems of identities see [[#References|[9]]]. A product of finitely-based varieties of groups is not necessarily finitely based; in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629055.png" /> has no finite basis.
+
A variety of groups is said to be finitely based if it can be specified by a given finite number of identities. These include, for example, all Cross, nilpotent and metabelian varieties. It has been proved [[#References|[8]]] that non-finitely based varieties of groups exist, and that the number of all varieties of groups has the power of the continuum. For examples of infinite independent systems of identities see [[#References|[9]]]. A product of finitely-based varieties of groups is not necessarily finitely based; in particular, $  \mathfrak B _ {4} \mathfrak B _ {2} $
 +
has no finite basis.
  
A variety of groups is a variety of Lie type if it is generated by its torsion-free nilpotent groups. If, in addition, the factors of the lower central series of the free groups of the variety are torsion-free groups, then the variety is said to be of Magnus type. The class of varieties of Lie type does not coincide with that of Magnus type; each of them is closed with respect to the operation of multiplication of varieties [[#References|[10]]]. Examples of varieties of Magnus type include the variety of all groups, the varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629057.png" />, and varieties obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629058.png" /> by the application of a finite number of operations of intersection and multiplication [[#References|[1]]].
+
A variety of groups is a variety of Lie type if it is generated by its torsion-free nilpotent groups. If, in addition, the factors of the lower central series of the free groups of the variety are torsion-free groups, then the variety is said to be of Magnus type. The class of varieties of Lie type does not coincide with that of Magnus type; each of them is closed with respect to the operation of multiplication of varieties [[#References|[10]]]. Examples of varieties of Magnus type include the variety of all groups, the varieties $  \mathfrak N _ {c} $,  
 +
$  \mathfrak A  ^ {n} $,  
 +
and varieties obtained from $  \mathfrak N _ {c} $
 +
by the application of a finite number of operations of intersection and multiplication [[#References|[1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Neumann,  "Varieties of groups" , Springer  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.I. Kargapolov,  V.A. Churkin,  "On varieties of solvable groups"  ''Algebra and Logic'' , '''10''' :  6  (1971)  pp. 359–398  ''Algebra i Logika'' , '''10''' :  6  (1971)  pp. 651–657</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.R.J. Groves,  "On varieties of solvable groups II"  ''Bull. Austr. Math. Soc.'' , '''7''' :  3  (1972)  pp. 437–441</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.A. Bryce,  "Metabelian groups and varieties"  ''Philos. Trans. Roy. Soc. London Ser. A'' , '''266'''  (1970)  pp. 281–355</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Brisley,  "Varieties of metabelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629059.png" />-groups of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629060.png" />"  ''J. Austr. Math. Soc.'' , '''12''' :  1  (1971)  pp. 53–62</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.Yu. Ol'shanskii,  "Solvable just-non-Cross varieties of groups"  ''Math. USSR Sb.'' , '''14''' :  1  (1971)  pp. 115–129  ''Mat. Sb.'' , '''85''' :  1  (1971)  pp. 115–131</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  Yu.P. Razmyslov,  "On Lie algebras satisfying the Engel condition"  ''Algebra and Logic'' , '''10''' :  1  (1971)  pp. 21–29  ''Algebra i Logika'' , '''10''' :  1  (1971)  pp. 33–44</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.Yu. Ol'shanskii,  "On the problem of a finite basis of identities in groups"  ''Math. USSR Izv.'' , '''4''' :  2  (1970)  pp. 381–389  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''34''' :  2  (1970)  pp. 376–384</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  S.I. Adyan,  "The Burnside problem and identities in groups" , Springer  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.L. Shmel'kin,  "Wreath product of Lie algebras and their applications in the theory of groups"  ''Proc. Moscow Math. Soc.'' , '''29'''  (1973)  pp. 239–252  ''Trudy Moskov. Mat. Obshch.'' , '''29'''  (1973)  pp. 247–260</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  Yu.M. Gorchakov,  "Commutator subgroups"  ''Sib. Math. J.'' , '''10''' :  5  (1969)  pp. 754–761  ''Sibirsk. Mat. Zh.'' , '''10''' :  5  (1969)  pp. 1023–1033</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Neumann,  "Varieties of groups" , Springer  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.I. Kargapolov,  V.A. Churkin,  "On varieties of solvable groups"  ''Algebra and Logic'' , '''10''' :  6  (1971)  pp. 359–398  ''Algebra i Logika'' , '''10''' :  6  (1971)  pp. 651–657</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.R.J. Groves,  "On varieties of solvable groups II"  ''Bull. Austr. Math. Soc.'' , '''7''' :  3  (1972)  pp. 437–441</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.A. Bryce,  "Metabelian groups and varieties"  ''Philos. Trans. Roy. Soc. London Ser. A'' , '''266'''  (1970)  pp. 281–355</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Brisley,  "Varieties of metabelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629059.png" />-groups of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096290/v09629060.png" />"  ''J. Austr. Math. Soc.'' , '''12''' :  1  (1971)  pp. 53–62</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.Yu. Ol'shanskii,  "Solvable just-non-Cross varieties of groups"  ''Math. USSR Sb.'' , '''14''' :  1  (1971)  pp. 115–129  ''Mat. Sb.'' , '''85''' :  1  (1971)  pp. 115–131</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  Yu.P. Razmyslov,  "On Lie algebras satisfying the Engel condition"  ''Algebra and Logic'' , '''10''' :  1  (1971)  pp. 21–29  ''Algebra i Logika'' , '''10''' :  1  (1971)  pp. 33–44</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.Yu. Ol'shanskii,  "On the problem of a finite basis of identities in groups"  ''Math. USSR Izv.'' , '''4''' :  2  (1970)  pp. 381–389  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''34''' :  2  (1970)  pp. 376–384</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  S.I. Adyan,  "The Burnside problem and identities in groups" , Springer  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.L. Shmel'kin,  "Wreath product of Lie algebras and their applications in the theory of groups"  ''Proc. Moscow Math. Soc.'' , '''29'''  (1973)  pp. 239–252  ''Trudy Moskov. Mat. Obshch.'' , '''29'''  (1973)  pp. 247–260</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  Yu.M. Gorchakov,  "Commutator subgroups"  ''Sib. Math. J.'' , '''10''' :  5  (1969)  pp. 754–761  ''Sibirsk. Mat. Zh.'' , '''10''' :  5  (1969)  pp. 1023–1033</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:28, 6 June 2020


A class of all groups satisfying a fixed system of identity relations, or laws,

$$ v ( x _ {1} \dots x _ {n} ) = 1, $$

where $ v $ runs through some set $ V $ of group words, i.e. elements of the free group $ X $ with free generators $ x _ {1} \dots x _ {n} , . . . $. Just like any variety of algebraic systems (cf. Algebraic systems, variety of), a variety of groups can also be defined by the property of being closed under subsystems (subgroups), homomorphic images and Cartesian products. The smallest variety containing a given class $ \mathfrak C $ of groups is denoted by $ \mathop{\rm var} \mathfrak C $. Regarding the operations of intersection and union of varieties, defined by the formula

$$ \mathfrak U \lor \mathfrak V = \mathop{\rm var} ( \mathfrak U \cup \mathfrak V ), $$

varieties of groups form a complete modular, but not distributive, lattice. The product $ \mathfrak U \mathfrak V $ of two varieties $ \mathfrak U $ and $ \mathfrak V $ is defined as the variety of groups consisting of all groups $ G $ with a normal subgroup $ N \in \mathfrak U $ such that $ G/N \in \mathfrak V $. Any variety of groups other than the variety of trivial groups and the variety of all groups can be uniquely represented as a product of varieties of groups which cannot be split further.

Examples of varieties of groups: the variety $ \mathfrak A $ of all Abelian groups; the Burnside variety $ \mathfrak B _ {n} $ of all groups of exponent (index) $ n $, defined by the identity $ x ^ {n} = 1 $; the variety $ \mathfrak A _ {n} = \mathfrak B _ {n} \wedge \mathfrak A $; the variety $ \mathfrak N _ {c} $ of all nilpotent groups of class $ \leq c $; the variety $ \mathfrak A ^ {l} $ of all solvable groups of length $ \leq l $; in particular, if $ l = 2 $, $ \mathfrak A ^ {2} $ is the variety of metabelian groups.

Let $ {\mathcal P} $ be some property of groups. One says that a variety of groups $ \mathfrak V $ has the property $ {\mathcal P} $( locally) if each (finitely-generated) group in $ \mathfrak V $ has the property $ {\mathcal P} $. One says, in this exact sense, that the variety is nilpotent, locally nilpotent, locally finite, etc.

The properties of a solvable variety of groups $ \mathfrak V $ depend on $ \mathfrak V \wedge \mathfrak A ^ {2} $. Thus, if $ \mathfrak B \supseteq \mathfrak A ^ {2} $, then $ \mathfrak V \subseteq \mathfrak B _ {n} \mathfrak N _ {c} \mathfrak B _ {n} $ for certain suitable $ n $ and $ c $[2], [3]. The description of metabelian varieties of groups is reduced, to a large extent, to the description of locally finite varieties of groups: If a metabelian variety $ \mathfrak V $ is not locally finite, then

$$ \mathfrak B = \ \mathfrak B _ {1} \lor \mathfrak B _ {2} \lor \mathfrak B _ {3} , $$

where $ \mathfrak B _ {1} = \mathfrak A _ {m} \mathfrak A $, $ \mathfrak V _ {2} $ is uniquely representable as the union of a finite number of varieties of groups of the form $ \mathfrak N _ {c} \mathfrak A _ {k} \wedge \mathfrak A ^ {2} $, and $ \mathfrak V _ {3} $ is locally finite [4]. Certain locally finite metabelian varieties have been described — for example, varieties of $ p $- groups of class $ \leq p + 1 $( cf. [5]).

A variety of groups is said to be a Cross variety if it is generated by a finite group. Cross varieties of groups are locally finite. A variety of groups is said to be a near Cross variety if it is not Cross, but each of its proper subvarieties is Cross. The solvable near Cross varieties are exhausted by the varieties $ \mathfrak A $, $ \mathfrak A _ {p} ^ {2} $, $ \mathfrak A _ {p} \mathfrak A _ {q} \mathfrak A _ {r} $, $ \mathfrak A _ {p} \mathfrak T _ {q} $, where $ p, q, r $ are different prime numbers, $ \mathfrak T _ {q} = \mathfrak B _ {q} \wedge \mathfrak N _ {2} $ for odd $ q $ and $ \mathfrak T _ {2} = \mathfrak B _ {4} \wedge \mathfrak N _ {2} $[6]. There exist, however, other near Cross varieties; such varieties are contained, for example, in any variety $ \mathfrak K $ of all locally finite groups of exponent $ p \geq 5 $[7]. An important role in the study of locally finite varieties of groups is played by critical groups — groups not contained in the variety generated by all their proper subgroups and quotient groups. A Cross variety can contain only a finite number of non-isomorphic critical groups. All locally finite varieties are generated by their critical groups.

A variety of groups is said to be finitely based if it can be specified by a given finite number of identities. These include, for example, all Cross, nilpotent and metabelian varieties. It has been proved [8] that non-finitely based varieties of groups exist, and that the number of all varieties of groups has the power of the continuum. For examples of infinite independent systems of identities see [9]. A product of finitely-based varieties of groups is not necessarily finitely based; in particular, $ \mathfrak B _ {4} \mathfrak B _ {2} $ has no finite basis.

A variety of groups is a variety of Lie type if it is generated by its torsion-free nilpotent groups. If, in addition, the factors of the lower central series of the free groups of the variety are torsion-free groups, then the variety is said to be of Magnus type. The class of varieties of Lie type does not coincide with that of Magnus type; each of them is closed with respect to the operation of multiplication of varieties [10]. Examples of varieties of Magnus type include the variety of all groups, the varieties $ \mathfrak N _ {c} $, $ \mathfrak A ^ {n} $, and varieties obtained from $ \mathfrak N _ {c} $ by the application of a finite number of operations of intersection and multiplication [1].

References

[1] H. Neumann, "Varieties of groups" , Springer (1967)
[2] M.I. Kargapolov, V.A. Churkin, "On varieties of solvable groups" Algebra and Logic , 10 : 6 (1971) pp. 359–398 Algebra i Logika , 10 : 6 (1971) pp. 651–657
[3] J.R.J. Groves, "On varieties of solvable groups II" Bull. Austr. Math. Soc. , 7 : 3 (1972) pp. 437–441
[4] R.A. Bryce, "Metabelian groups and varieties" Philos. Trans. Roy. Soc. London Ser. A , 266 (1970) pp. 281–355
[5] W. Brisley, "Varieties of metabelian -groups of class " J. Austr. Math. Soc. , 12 : 1 (1971) pp. 53–62
[6] A.Yu. Ol'shanskii, "Solvable just-non-Cross varieties of groups" Math. USSR Sb. , 14 : 1 (1971) pp. 115–129 Mat. Sb. , 85 : 1 (1971) pp. 115–131
[7] Yu.P. Razmyslov, "On Lie algebras satisfying the Engel condition" Algebra and Logic , 10 : 1 (1971) pp. 21–29 Algebra i Logika , 10 : 1 (1971) pp. 33–44
[8] A.Yu. Ol'shanskii, "On the problem of a finite basis of identities in groups" Math. USSR Izv. , 4 : 2 (1970) pp. 381–389 Izv. Akad. Nauk SSSR Ser. Mat. , 34 : 2 (1970) pp. 376–384
[9] S.I. Adyan, "The Burnside problem and identities in groups" , Springer (1979) (Translated from Russian)
[10] A.L. Shmel'kin, "Wreath product of Lie algebras and their applications in the theory of groups" Proc. Moscow Math. Soc. , 29 (1973) pp. 239–252 Trudy Moskov. Mat. Obshch. , 29 (1973) pp. 247–260
[11] Yu.M. Gorchakov, "Commutator subgroups" Sib. Math. J. , 10 : 5 (1969) pp. 754–761 Sibirsk. Mat. Zh. , 10 : 5 (1969) pp. 1023–1033

Comments

The Oates–Powell theorem says that the variety generated by the finite groups is Cross. As a corollary it follows that the identities of finite groups admit a finite basis.

In [a1] the concept of varieties for a large class of algebraic structures was brought forward. The first systematic study of varieties of groups is [a2].

References

[a1] G. Birkhoff, "On the structure of abstract algebras" Proc. Cambridge Phil. Soc. , 31 (1935) pp. 433–454
[a2] B.H. Neumann, "Identical relations in groups I" Math. Ann. , 114 (1937) pp. 506–525
How to Cite This Entry:
Variety of groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Variety_of_groups&oldid=49129
This article was adapted from an original article by A.L. Shmel'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article