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FC-groups. (English) Zbl 0547.20031

An FC-group is a group in which each element has only a finite number of conjugates. There is now a substantial body of knowledge about these groups, many of the more recent developments being due to the present author and to the Russian mathematician Gorchakov. These notes give a comprehensive account of most of the more important aspects of the structure theory of FC-groups. These groups have also become important in the algebraic study of infinite group rings, where various normal FC- subgroups of a group strongly influence the ideal structure of its group rings, for example. This aspect is not touched on in these notes, which deal with purely group theoretic matters. A good account of the role of FC-groups in group rings can be found in D. S. Passman’s book ”The algebraic structure of group rings” (1977; Zbl 0368.16003). Another account of group theoretic properties of FC-groups is given in Yu. M. Gorchakov’s book ”Groups with finite classes of conjugate elements” (1978; Zbl 0496.20025), which is only available in Russian. The notes under review bring some order and unification into the subject by giving prominence to certain important results or points of view, for example, Gorchakov’s result on periodic FC-groups that are residually of bounded order (see below) is emphasized, and the profinite structure of the group of locally inner automorphisms is brought to the fore. Further lines of development are suggested by a liberal sprinkling of open problems.
Since the commutator group and the central quotient group of an FC-group are both periodic, much of the attention in the study of FC-groups focuses on the periodic case. It is easy to prove that every homomorphic image of a subgroup of a restricted direct product of finite groups is a periodic FC-group, and every subgroup of a restricted direct product of finite groups is a residually finite periodic FC-group. Chapters 2 and 3 of the notes deal with partial converses to these observations. In a basic paper [J. Lond. Math. Soc. 34, 289-304 (1959; Zbl 0088.022)] P. Hall showed that the converses of both statements are true for countable groups, but this is not so for uncountable groups in general. Much of Tomkinson’s treatment of residually finite periodic FC-groups is based on an important theorem of Gorchakov to the effect that if such a group can be embedded in a cartesian product of finite groups of bounded order (so it is ”residually of bounded order”) then it can be embedded in a restricted direct product of finite groups. This can be viewed as a non commutative analogue of the fact that an abelian group of finite exponent is a restricted direct product of finite cyclic groups. It enables residually finite periodic FC-groups to be characterized as subgroups of ”centrally restricted” products of finite groups where a centrally restricted product of finite groups is just the subgroup of the cartesian product generated by the restricted direct product and the torsion subgroup of the centre. From this, many other results relating residually finite periodic FC-groups to subgroups of restricted direct products of finite groups, are deduced. Thus a pleasing unification of a number of previously disparate results is obtained.
Chapter 3 studies the relation between periodic FC-groups and homomorphic images of subgroups of restricted direct products of finite groups \((QSD{\mathfrak F}\)-groups for short). The situation here is less clear-cut. Classes \({\mathfrak Y}\) and \({\mathfrak Z}\) are defined as follows. We have \(G\in {\mathfrak Z}\) if and only if, for each infinite cardinal \({\mathfrak m}\) and subset \(S\subseteq G\) with \(| S| <{\mathfrak m}\), we have \(| G:C_ G(S)| <{\mathfrak m}\). We also say \(G\in {\mathfrak Y}\) if and only if G is periodic and for each infinite cardinal \({\mathfrak m}\) and subgroup \(H\leq G\) with \(| H| <{\mathfrak m}\), we have \(| G:N_ G(H)| <{\mathfrak m}\). One easily sees that \[ QSD{\mathfrak F}\subseteq\{\text{periodic \({\mathfrak Z}\)-groups})\subseteq {\mathfrak Y}\subseteq\{\text{periodic FC-groups}\}. \] It is apparently unknown whether the first three classes are distinct; however we have that a periodic FC-group belongs to \({\mathfrak Y}\) if and only if each of its extraspecial sections belongs to \({\mathfrak Y}\). This focuses attention on extraspecial groups. Strange examples of these groups are constructed from bizarre symplectic forms on vector spaces of uncountable dimension over GF[p]. Here, set theory comes into play. There appear to be a number of interesting open questions, and some are stated.
Chapter 4 discusses locally inner automorphisms. A locally inner automorphism is one which, on any given finite subset, agrees with some inner automorphism. In the theory of FC-groups, these take over the part played by inner automorphisms in finite group theory. The group Linn G of locally inner automorphisms of an FC-group G has a natural structure of a profinite group, and though no deep properties of profinite groups are brought into play, this provides a useful point of view. The following result, based on an argument of D. Holt, enables Tomkinson to give a simplified and improved treatment of certain results from the literature on local conjugacy classes of subgroups. Let L be a profinite group and \(\{K_ i:\) \(i\in I\}\) be a system of open normal subgroups of L such that the intersection of any two contains a third, and \(\cap_{i\in I}K_ i=1\). Let H be a closed subgroup of infinite index in L and \(\alpha\) be the cardinality of the set of distinct subgroups \(KH_ i\). Then \(| L:H| =\exp \alpha.\)
Chapters 5 and 6 are devoted to Sylow theory, formations and Fitting classes. On the whole, one can say that results concerning Sylow subgroups, Hall subgroups, projectors, injectors and the like, generalize well to periodic FC-groups, provided one replaces ordinary conjugacy by conjugacy under the group of locally inner automorphisms (local conjugacy). A detailed treatment of these results is given.
Chapter 7 deals with applications of combinatorial set theory in the theory of FC-groups. For example we have Theorem 7.17 (i) A group G is finite-by-abelian if and only if \(| A^ G:A|\) is finite for every abelian subgroup A of G. (ii) If \({\mathfrak m}\) is an infinite cardinal and G an FC-group, then \(| G'| <{\mathfrak m}\) if and only if \(| U^ G:U| <{\mathfrak m}\) for every subgroup U of G.
The final chapter contains miscellaneous topics, including a discussion of groups with finite layers, and of minimal non FC-groups, about which there are a number of open questions.
There is some parallel between periodic FC-groups, which are exactly groups generated by finite normal subgroups, and Lie algebras generated by finite dimensional ideals. For an account of the latter, see I. N. Stewart ”Lie algebras generated by finite dimensional ideals” (1975; Zbl 0325.17002).
Reviewer: B.Hartley

MSC:

20F24 FC-groups and their generalizations
20F50 Periodic groups; locally finite groups
20E34 General structure theorems for groups
20-02 Research exposition (monographs, survey articles) pertaining to group theory
20E07 Subgroup theorems; subgroup growth
20E36 Automorphisms of infinite groups
20F17 Formations of groups, Fitting classes