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 $\mathop{\rm var} \mathfrak C$. 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