On finite simple and nonsolvable groups acting on homology 4-spheres

The only finite non-Abelian simple group acting on a homology 3-sphere—necessarily non-freely—is the dodecahedral group A₅≅PSL(2,5) (in analogy, the only finite perfect group acting freely on a homology 3-sphere is the binary dodecahedral group ). In the present paper we show that the only finite simple groups acting on a homology 4-sphere, and in particular on the 4-sphere, are the alternating or linear fractional groups A₅≅PSL(2,5) and A₆≅PSL(2,9). From this we deduce a short list of groups which contains all finite nonsolvable groups admitting an action on a homology 4-sphere.     

Finite Groups Acting on Homology 3-Spheres: On a Result of M. Reni

 We give a list including all finite groups G which admit smooth orientation preserving actions on homology 3-spheres (arbitrary actions, i.e. possibly with fixed points; if the action is free then the group G has periodic cohomology and the classification of such groups is well known). The main work in this direction is due to M. Reni. In the present paper, we complete and extend his results for the case of nonsolvable groups G. Received 19 March 2001; in revised form 15 September 2001     

Fixity and free group actions on products of spheres

A representation G U(n) of degree n has fixity equal to the smallest integer f such that the induced action of G on U(n) /U(n-f-1) is free. Using bundle theory we show that if G admits a representation of fixity one, then it acts freely and smoothly on We use this to prove that a finite p-group (for p > 3) acts freely and smoothly on a product of two spheres if and only if it does not contain ( /p)³ as a subgroup. We use propagation methods from surgery theory to show that a representation of fixity f < n - 1 gives rise to a free action of G on a product of f + 1 spheres provided the order of G is relatively prime to (n - 1)!. We give an infinite collection of new examples of finite p-groups of rank r which act freely on a product of r spheres, hence verifying a strong form of a well-known conjecture for these groups. In addition we show that groups of fixity two act freely on a finite complex with the homotopy type of a product of three spheres. A number of examples are explicitly described.     

On products of two nilpotent subgroups of a finite group

LetG be a finite group with an abelian Sylow 2-subgroup. LetA be a nilpotent subgroup ofG of maximal order satisfying class (A)≦k, wherek is a fixed integer larger than 1. Suppose thatA normalizes a nilpotent subgroupB ofG of odd order. ThenAB is nilpotent. Consequently, ifF(G) is of odd order andA is a nilpotent subgroup ofG of maximal order, thenF(G)?A.     

Finite Groups in Which Sylow Normalizers Admit Supplements

For a group G and a subgroup M of G, we say that a subgroup A of G is a supplement to M in G, if G = MA. We prove that a finite group in which every Sylow normalizer admits a nilpotent supplement is solvable. In particular we confirm a conjecture of Buchthal. Moreover we investigate finite groups in which Sylow normalizers admit solvable supplements. Supported by SRFDP(20010001046)     

Topologically equivalent N-dimensional isometries

Let G be the group of isometries of the n-sphere, Euclidean n-space, or hyperbolic n-space, the group of similarities of Euclidean n-space, or the group of Möbius transformations of the n-sphere. In each case we attempt to determine the conjugacy classes in G which are amalgamated when we allow conjugation of the elements of G by homeomorphisms of the space on which G acts. We are successful modulo undetermined amalgamation among certain periodic orthogonal transformations.     

On a Fitting length conjecture without the coprimeness condition

Let A be a finite nilpotent group acting fixed point freely by automorphisms on the finite solvable group G. It is conjectured that the Fitting length of G is bounded by the number of primes dividing the order of A, counted with multiplicities. The main result of this paper shows that the conjecture is true in the case where A is cyclic of order p ⁿ q, for prime numbers p and q coprime to 6 and G has abelian Sylow 2-subgroups.     

Class-Preserving Coleman Automorphisms of Finite Groups

 Let G be a finite group whose Sylow 2-subgroups are either cyclic, dihedral, or generalized quaternion. It is shown that a class-preserving automorphism of G of order a power of 2 whose restriction to any Sylow subgroup of G equals the restriction of some inner automorphism of G is necessarily an inner automorphism. Interest in such automorphisms arose from the study of the isomorphism problem for integral group rings, see [6, 7, 13, 14]. Received 30 September 2001; in revised form 10 December 2001     

On infinite tournaments with regular automorphism groups

Atournament regular representation (TRR) of an abstract groupG is a tournamentT whose automorphism group is isomorphic toG and is a regular permutation group on the vertices ofT. L. Babai and W. Imrich have shown that every finite group of odd order exceptZ ₃ ×Z ₃ admits a TRR. In the present paper we give several sufficient conditions for an infinite groupG with no element of order 2 to admit a TRR. Among these are the following: (1)G is a cyclic extension byZ of a finitely generated group; (2)G is a cyclic extension byZ _2n+1 of any group admitting a TRR; (3)G is a finitely generated abelian group; (4)G is a countably generated abelian group whose torsion subgroup is finite.     

On SS-Quasinormal Subgroups of Finite Groups

A subgroup of a group G is said to be Sylow-quasinormal (S-quasinormal) in G if it permutes with every Sylow subgroup of G. A subgroup H of a group G is said to be Supplement-Sylow-quasinormal (SS-quasinormal) in G if there is a supplement B of H to G such that H is permutable with every Sylow subgroup of B. In this article, we investigate the influence of SS-quasinormal of maximal or minimal subgroups of Sylow subgroups of the generalized Fitting subgroup of a finite group.     

On the order of the unitary subgroup of a modular group algebra

Let KG be a group algebra of a finite p-group G over a finite field Kof characteristic p. We compute the order of the unitary subgroup of the group of units when G is either an extraspecial 2-group or the central product of such a group with a cyclic group of order 4 or G has an abelian subgroup A of index 2 and an element b such that b inverts each element of A.     

Topologically equivalent two-dimensional isometries

Let G be the group of isometries of the 2-sphere, the Euclidean plane or the hyperbolic plane, the group of similarities of the Euclidean plane or the group of Möbius transformations of the 2-sphere. In each instance we determine which conjugacy classes in G are amalgamated when we allow conjugation of the elements of G by homeomorphisms of the space on which G acts. Our results are related to recent work on the homeomeric classification of two-dimensional patterns.     

Finite groups with <Emphasis Type="Italic">S</Emphasis>-supplemented <Emphasis Type="Italic">p</Emphasis>-subgroups

Consider a finite group G. A subgroup is called S-quasinormal whenever it permutes with all Sylow subgroups of G. Denote by B _sG the largest S-quasinormal subgroup of G lying in B. A subgroup B is called S-supplemented in G whenever there is a subgroup T with G = BT and B∩T ≤ B _sG . A subgroup L of G is called a quaternionic subgroup whenever G has a section A/B isomorphic to the order 8 quaternion group such that L ≤ A and L ∩ B = 1. This article is devoted to proving the following theorem.     

Class-Preserving Coleman Automorphisms of Finite Groups

 Let G be a finite group whose Sylow 2-subgroups are either cyclic, dihedral, or generalized quaternion. It is shown that a class-preserving automorphism of G of order a power of 2 whose restriction to any Sylow subgroup of G equals the restriction of some inner automorphism of G is necessarily an inner automorphism. Interest in such automorphisms arose from the study of the isomorphism problem for integral group rings, see [6, 7, 13, 14].     

Hyperbolic isometries versus symmetries of links

We prove that every finite group is the orientation-preserving isometry group of the complement of a hyperbolic link in the 3-sphere.     

Some Notes on Meet-Irreducible Subgroups of Finite Groups

A proper subgroup A of a finite group G is said to be primitive or meet-irreducible if there is a unique subgroup A₀ ≤ G such that A is a maximal subgroup of A₀. In this case we say that |A₀: A| is the small index of A and denote it by |G: A|₀. In this article, we study the influence of meet-irreducible subgroups and their small indexes on the structure of G. In particular, we prove that a finite group G is supersoluble if and only if |G: A|₀ = |G: B|₀ for any two meet-irreducible subgroups A and B of G with A_G = B_G.     

On some infinite dimensional linear groups

Let F be a field, A be a vector space over F, and GL(F,A) the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dim _F (BFG/B) is finite. A subspace B is called almost G-invariant, if dim _F (B/Core _G (B)) is finite. In the present article we begin the study of subgroups G of GL(F,A) such that every subspace of A is either nearly G-invariant or almost G-invariant. More precisely, we consider the case when G is a periodic p′-group where p = charF.      

Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold

The paper studies the first homology of finite regular branched coverings of a universal Borromean orbifold called B _4,4,4ℍ³. We investigate the irreducible components of the first homology as a representation space of the finite covering transformation group G. This gives information on the first betti number of finite coverings of general 3-manifolds by the universality of B _4,4,4. The main result of the paper is a criterion in terms of the irreducible character whether a given irreducible representation of G is an irreducible component of the first homology when G admits certain symmetries. As a special case of the motivating argument the criterion is applied to principal congruence subgroups of B _4,4,4. The group theoretic computation shows that most of the, possibly nonprincipal, congruence subgroups are of positive first Betti number. This work is partially supported by the Sonderforschungsbereich 288.     

Mod 3 cohomology algebras of finite H-spaces

Let X be a 3 local, finite, simply connected H-space with associative homology ring . Some known examples are the Lie group , Harper's H-space X(3) and any odd dimensional sphere . We prove the cohomology algebra is isomorphic to the cohomology algebra of a finite product of and odd dimensional spheres. Received: 15 May 2001; in final form: 22 May 2001 / Published online: 28 February 2002