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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|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

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

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 $[[#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

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

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 [[#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  $  \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,  $  \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  $  \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====
<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====
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 [[#References|[a1]]] the concept of varieties for a large class of algebraic structures was brought forward. The first systematic study of varieties of groups is [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Birkhoff,   "On the structure of abstract algebras"  ''Proc. Cambridge Phil. Soc.'' , '''31'''  (1935)  pp. 433–454</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.H. Neumann,   "Identical relations in groups I"  ''Math. Ann.'' , '''114'''  (1937)  pp. 506–525</TD></TR></table>