Variety of groups

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 $\mathrm{var}\mathfrak{C}$. Regarding the operations of intersection and union of varieties, defined by the formula

$$\mathfrak{U}\vee \mathfrak{V}=\mathrm{var}(\mathfrak{U}\cup \mathfrak{V}),$$

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 $\le c$; the variety ${\mathfrak{A}}^l$ of all solvable groups of length $\le l$; 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

$$\mathfrak{B}={\mathfrak{B}}_1\vee {\mathfrak{B}}_2\vee {\mathfrak{B}}_3,$$

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 $\le p+1$( 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 $p\ge 5$[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[edit]

[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[edit]

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[edit]