# Group with a finiteness condition

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A group whose elements or subgroups satisfy some finiteness condition. A finiteness condition in group theory is understood to mean any property of all finite groups (cf. Finite group) such that there exist infinite groups not having this property. In group-theoretic studies the following finiteness conditions are the most important: finiteness of decreasing chains of subgroups (the minimum condition for subgroups, cf. Artinian group), finiteness of ascending chains of subgroups (the maximum condition for subgroups, cf. Noetherian group), being finitely generated (cf. Finitely-generated group), finiteness of the orders of elements (periodicity, cf. Periodic group), finiteness of finitely-generated subgroups (local finiteness, cf. Locally finite group), finiteness of rank (cf. Rank of a group), and finiteness of conjugacy classes (cf. Conjugate elements).

Systematic studies of groups with a finiteness condition began in 1939–1940 [1] with the study of locally nilpotent and locally solvable groups with the minimum condition for subgroups (cf. also Locally nilpotent group; Locally solvable group), as a result of which it was established that infinite groups of this type are finite extensions of direct products of a finite number of quasi-cyclic groups (cf. Quasi-cyclic group). It is not yet (1977) known if this theorem is also valid for an arbitrary infinite group with the minimum condition for subgroups. If local finiteness is assumed, the answer to the problem is positive [5]. The study of groups with the minimum condition has enriched group theory with important results. Substantial restrictions were then imposed on the minimum condition itself, by imposing it not on all subgroups, but only on those satisfying certain additional special requirements (such as normality or commutativity, or having finite index or prime-power order). The study of groups with the maximum condition proved to be less fruitful than that of groups with the minimum condition. Solvable groups (cf. Solvable group) with the maximum condition are polycyclic (cf. Polycyclic group). In solvable groups the maximum condition for subgroups is equivalent to the maximum condition for Abelian subgroups [4]. A similar result was also established for the minimum condition in locally solvable groups. The maximum condition for subgroups is equivalent to the fact that the group and all its subgroups are finitely generated. For nilpotent groups (cf. Nilpotent group) it is equivalent to the group itself being finitely generated.

A group is of finite rank if the minimum number of generating elements in each of its finitely-generated subgroups does not exceed a certain fixed number. This finiteness condition was widely used in studies of solvable groups and locally nilpotent groups. It was found, in particular, that if all Abelian subgroups of a locally nilpotent torsion-free group (cf. Group without torsion) are of finite rank, then the group itself is also of finite rank [4].

Studies of groups with all conjugacy classes finite yielded several important results. The most thoroughly studied are layer-finite groups, i.e. groups with finite sets of elements of each order. Their study in fact resulted in a complete description of their structure. It follows from this description, in particular, that the class of layer-finite groups is identical with the class of locally normal groups (cf. Locally normal group) satisfying the minimum condition for primary subgroups.

#### References

 [1] S.N. Chernikov, "Finiteness conditions in general group theory" Uspekhi Mat. Nauk , 14 : 5(89) (1959) pp. 45–96 (In Russian) [2] D.J.S. Robinson, "Finiteness condition and generalized soluble groups" , 1–2 , Springer (1972) [3] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) [4] A.I. Mal'tsev, "On some classes of infinite solvable groups" Mat. Sb. , 28 : 3(70) (1951) pp. 567–588 (In Russian) [5] V.P. Shunkov, "On locally finite groups with a minimality condition for Abelian subgroups" Algebra and Logic , 9 : 5 (1970) pp. 350–370 Algebra i Logika , 9 : 5 (1970) pp. 579–615