Martin R. Pettet

Three-term relations for Hardy sums

Abstract L. A. Goldberg (thesis, Univ. of Illinois, Urbana, 1981) discovered some three-term and mixed three-term relations for Hardy sums. His proofs are based on Berndts transformation formulae for the logarithms of the classical theta functions. In this paper, we give elementary proofs for all of Goldbergs results and also prove some new three-term relations for Dedekind sums.

The automorphism group of a graph product of groups

A theorem of Karrass, Pietrowski and Solitar on the structure of the automorphism group of an amalgamated free product is extended to automorphism groups of fundamental groups of graphs of groups in which the edge groups are incomparable up to conjugacy.

Automorphisms and Fitting factors of finite groups

It is proved that for every prime p, there exists a function fp such that, if G is a finite solvable group with an automorphism α of order p, then G contains a nilpotent normal subgroup of index at most fp(¦CG(α)¦). The question is first reduced to a representation theoretic problem which is then analyzed along the lines of the well-known Theorem B of Hall and Higman. Using an elementary argument of Brauer and Fowler, it is shown further that in the case p = 2, the assumption of solvability may be omitted. Thus, there is a function f such that for any group G of even order and any involution x in G, the Fitting factor group GF(G) has order at most f(¦CG(x)¦).

Virtually free groups with finitely many outer automorphisms

Let G be a finitely generated virtually free group. From a presentation of G as the fundamental group of a finite graph of finite-by-cyclic groups, necessary and sufficient conditions are derived for the outer automorphism group of G to be finite. Two versions of the characterization are given, both effectively verifiable from the graph of groups. The more purely group theoretical criterion is expressed in terms of the structure of the normalizers of the edge groups (Theorem 5.10); the other version involves certain finiteness conditions on the associated G-tree (Theorem 5.16). Coupled with an earlier result, this completes a description of the finitely generated groups whose full automorphism groups are virtually free.

On inner automorphisms of finite groups

It is shown that in certain classes of finite groups, inner automorphisms are characterized by an extension property and also by a dual lifting property. This is a consequence of the fact that for any finite group G and any prime p, there is a p-group P and a semidirect product H = GP such that P is characteristic in H and every automorphism of H induces an inner automorphism on H/P. Every inner automorphism of a group G possesses the property that it extends to an automorphism of any group which contains G as a subgroup. P. Schupp [2] has shown that in the category of all groups, this extendibility property characterizes inner automorphisms. However, since Schupps argument uses a free product construction, the question arises whether inner automorphisms can be similarly characterized in classes of groups which are not closed under free products. In this note, we confirm that the extendibility property does indeed characterize inner automorphisms in certain classes of finite groups. As it happens, with the same stroke the dual question of whether inner automorphisms are precisely those which can be lifted to homomorphic pre-images is also settled (affirmatively) for those classes. The basis for these conclusions amounts to an observation about a remarkable construction of H. Heineken and H. Liebeck [1] (and a subsequent extension of U. H. M. Webb [3]). Theorem. Let G be a finite group and let p be any prime. Then there exists a finite special p-group P on which G actsfaithfully such that P is a characteristic subgroup of the corresponding semidirect product H = GP, and such that every automorphism of H induces on H/P an inner automorphism. Corollary. Let 7r be any set of primes and let F be the class of all finite 7rgroups, the class of all finite solvable it-groups, or the class of all finite nilpotent 7t-groups. Suppose that G E F and that a is an automorphism of G. Then the following statements are equivalent: (i) a is an inner automorphism of G. Received by the editors September 14, 1988 and, in revised form, November 16, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 20D45. (? 1989 American Mathematical Society 0002-9939/89

Unfused Involutions in Finite Groups

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Distributively generated gc near-rings

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A note on the automorphism tower theorem for finite groups

ABSTRACT Let x be a p-element of a finite group G. We say that x is unfused in G if, for some Sylow p-subgroup S of G containing x, all G-conjugates of x in S are S-conjugates. It is shown (using the classification of finite simple groups) that a finite group that contains an unfused involution has a chief factor of order 2. #Communicated by M. Dixon.

Unfused Involutions in Finite Groups: An Addendum

Let N be a finite GC near-ring. Those near-rings N such that N is distributilvely generated and (N,+) is solvable are shown to be a direct sum of fields and d.g. ” basic near-rings of size 2“. These basic near-rings of size 2 are characterized. A method for constructing d.g. GC near-rings is presented. This work gives rise to a class of d.g. GC near-rings which are not centralizer near-rings.

Near-fields and linear transformations of finite fields

If G is a group with trivial center, the automorphism tower of G is the series G = Ao <Al <A2 < ., * where Ai+ l is the automorphism group of A, (and A, is identified with its inner automorphisms). It is a theorem of Wielandt [5] that if G is finite, its automorphism tower has finite height. In fact, slightly more generally, Wielandt showed that if G is a finite subnormal subgroup of A and CA(G) = 1, then