On the Loewy length of the center of a block with elementary abelian defect groups

Let G be a finite group and k an algebraically closed field of characteristic p. In this paper we investigate the Loewy structure of centers of indecomposable group algebras kG, for groups G with a normal elementary abelian Sylow p-subgroup. Furthermore, we show a reduction result for the case that a normal abelian Sylow p-subgroup is acted upon by a subgroup of its automorphism group; this is fundamental in providing generic formulae for the Loewy lengths considered.     

Some Hadamard designs with parameters (71,35,17)

Up to isomorphisms there are precisely eight symmetric designs with parameters (71, 35, 17) admitting a faithful action of a Frobenius group of order 21 in such a way that an element of order 3 fixes precisely 11 points. Five of these designs have 84 and three have 420 as the order of the full automorphism group G. If |G| = 420, then the structure of G is unique and we have G = (Frob₂₁ × Z5):Z4. In this case Z(G) = 〈1〉, G′ has order 35, and G induces an automorphism group of order 6 of Z7. If |G| = 84, then Z(G) is of order 2, and in precisely one case a Sylow 2‐subgroup is elementary abelian. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 144–149, 2002; DOI 10.1002/jcd.996     

Some new results on 1‐rotational 2‐factorizations of the complete graph

It is known that a necessary condition for the existence of a 1‐rotational 2‐factorization of the complete graph K_2n+1 under the action of a group G of order 2n is that the involutions of G are pairwise conjugate. Is this condition also sufficient? The complete answer is still unknown. Adapting the composition technique shown in Buratti and Rinaldi, J Combin Des, 16 (2008), 87–100, we give a positive answer for new classes of groups; for example, the groups G whose involutions lie in the same conjugacy class and having a normal subgroup whose order is the greatest odd divisor of |G|. In particular, every group of order 4t+2 gives a positive answer. Finally, we show that such a composition technique provides 2‐factorizations with a rich group of automorphisms. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 237–247, 2010     

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].     

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     

Finite Groups with Some Subgroups of Prime Power Order S-Quasinormally Embedded

Abstract

A subgroup of a group G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. A subgroup H of a group G is said to be S-quasinormally embedded in G if every Sylow subgroup of H is a Sylow subgroup of some S-quasinormal subgroup of G. In this paper we examine the structure of a finite group G under the assumption that certain abelian subgroups of prime power order are S-quasinormally embedded in G. Our results improve and extend recent results of Ramadan [Ramadan, M. (2001). The influence of S-quasinormality of some subgroups of prime power order on the structure of finite groups. Arch. Math. 77:143–148].     

Finite Groups Whose Generalized Hypercenter Contains Some Subgroups of Prime Power Order

A subgroup of a group G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. A subgroup H of a group G is said to be S-quasinormally embedded in G if every Sylow subgroup of H is a Sylow subgroup of some S-quasinormal subgroup of G. In this article, we investigate the structure of the finite group G under the assumption that certain abelian subgroups of prime power order are S-quasinormally embedded in G and lie in the generlized hypercenter of G.     

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.     

Structure of augmentation quotients of finite homocyclic abelian groups

Let G be a finite abelian group and its Sylow p-subgroup a direct product of copies of a cyclic group of order p~r,i.e.,a finite homocyclic abelian group.LetΔ~n (G) denote the n-th power of the augmentation idealΔ(G) of the integral group ring ZG.The paper gives an explicit structure of the consecutive quotient group Q_n(G)=Δ~n(G)/Δ~(n 1)(G) for any natural number n and as a consequence settles a problem of Karpilovsky for this particular class of finite abelian groups.     

Coleman automorphisms of finite groups

An automorphism of a finite group G whose restriction to any Sylow subgroup equals the restriction of some inner automorphism of G shall be called Coleman automorphism, named for D. B. Coleman, who's important observation from [2] especially shows that such automorphisms occur naturally in the study of the normalizer of G in the units of the integral group . Let Out be the image of these automorphisms in Out. We prove that Out is always an abelian group (based on previous work of E. C. Dade, who showed that Out is always nilpotent). We prove that if no composition factor of G has order p (a fixed prime), then Out is a -group. If O, it suffices to assume that no chief factor of G has order p. If G is solvable and no chief factor of has order 2, then , where is the center of . This improves an earlier result of S. Jackowski and Z. Marciniak. Received: 26 May 2000; in final form: 5 October 2000 / Published online: 19 October 2001     

Finite groups with trivial Frattini subgroup

All groups considered are finite. A group has a trivial Frattini subgroup if and only if every nontrivial normal subgroup has a proper supplement.The property is normal subgroup closed, but neither subgroup nor quotient closed. It is subgroup closed if and only if the group is elementary, i.e. all Sylow subgroups are elementary abelian. If G is solvable, then G and all its quotients have trivial Frattini subgroup if and only if every normal subgroup of G has a complement. For a nilpotent group, every nontrivial normal subgroup has a supplement if and only if the group is elementary abelian. Consequently, the center of a group in which every normal subgroup has a supplement is an elementary abelian direct factor.     

On the abelianizations of congruence subgroups of Aut(<Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>)

Let F _n be the free group of rank n, and let Aut⁺(F _n ) be its special automorphism group. For an epimorphism π : F _n → G of the free group F _n onto a finite group G we call the standard congruence subgroup of Aut⁺(F _n ) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ⁺(G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ⁺(G, π) ≤ Aut⁺(F ₂) has infinite abelianization.     

Constructions for Terraces and R‐Sequencings,Including a Proof That Bailey's Conjecture Holds for Abelian Groups

Every abelian group of even order with a noncyclic Sylow 2‐subgroup is known to be R‐sequenceable except possibly when the Sylow 2‐subgroup has order 8. We construct an R‐sequencing for many groups with elementary abelian Sylow 2‐subgroups of order 8 and use this to show that all such groups of order other than 8 also have terraces. This completes the proof of Bailey's Conjecture in the abelian case: all abelian groups other than the noncyclic elementary abelian 2‐groups have terraces. For odd orders it is known that abelian groups are R‐sequenceable except possibly those with noncyclic Sylow 3‐subgroups. We show how the theory of narcissistic terraces can be exploited to find R‐sequencings for many such groups, including infinitely many groups with each possible of Sylow 3‐subgroup type of exponent at most 3¹² and all groups whose Sylow 3‐subgroups are of the form or .     

Characterization of 3-Rewritable Finite Nilpotent Groups

Let n be an integer greater than 1. A group G is said to be “ n-rewritable” whenever for every n elements x ₁,…,x _n of G, there exist distinct permutations τ, σ on the set {1,2,…, n} such that x _τ(1) ··· x _τ(n) = x _{σ (1)} ··· x _{σ (n)}. In this article, we complete the classification of 3-rewritable finite nilpotent groups and prove that a finite nilpotent group G is 3-rewritable if and only if G has an abelian subgroup of index 2 or the derived subgroup has order < 6.     

Gorenstein Projective,Injective, and Flat Complexes

Abstract

We study the classification of those finite groups G having a non-inner class preserving automorphism. Criteria for these automorphisms to be inner are established. Let G be a nilpotent-by-nilpotent group and S?∈?Sy l ₂(G). If S is abelian, generalized quaternion or S is dihedral, and in this case G is also metabelian, then Out _c (G)?=?1. If S is generalized quaternion, 𝒵(S)???𝒵(G) and S ₄ is not a homomorphic image of G, then Out _c (G)?=?1. As a consequence, it follows that the normalizer problem of group rings has a positive answer for these groups.     

π-quasinormally embedded and c-supplemented subgroup of finite group

Suppose that H is a subgroup of a finite group G. H is called π-quasinormal in G if it permutes with every Sylow subgroup of G; H is called π-quasinormally embedded in G provided every Sylow subgroup of H is a Sylow subgroup of some π-quasinormal subgroup of G; H is called c-supplemented in G if there exists a subgroup N of G such that G = HN and H ∩ N ⩽ H _G = Core _G (H). In this paper, finite groups G satisfying the condition that some kinds of subgroups of G are either π-quasinormally embedded or c-supplemented in G, are investigated, and theorems which unify some recent results are given.      

On two classes of finite supersoluble groups

Let ? be a complete set of Sylow subgroups of a finite group G, that is, a set composed of a Sylow p-subgroup of G for each p dividing the order of G. A subgroup H of G is called ?-S-semipermutable if H permutes with every Sylow p-subgroup of G in ? for all p?π(H); H is said to be ?-S-seminormal if it is normalized by every Sylow p-subgroup of G in ? for all p?π(H). The main aim of this paper is to characterize the ?-MS-groups, or groups G in which the maximal subgroups of every Sylow subgroup in ? are ?-S-semipermutable in G and the ?-MSN-groups, or groups in which the maximal subgroups of every Sylow subgroup in ? are ?-S-seminormal in G.     

On the Artin exponent of some rational groups

A finite group G is called rational if all its irreducible complex characters are rational valued. In this paper, we show that if G is a direct product of finitely many rational Frobenius groups then every rationally represented character of G is a generalized permutation character. Also we show that the same assertion holds when G is a solvable rational group with a Sylow 2-subgroup isomorphic to the dihedral group of order 8 and an abelian normal Sylow 3-subgroup.     

One‐factorizations of complete graphs with vertex‐regular automorphism groups

We consider one‐factorizations of K_2n possessing an automorphism group acting regularly (sharply transitively) on vertices. We present some upper bounds on the number of one‐factors which are fixed by the group; further information is obtained when equality holds in these bounds. The case where the group is dihedral is studied in some detail, with some non‐existence statements in case the number of fixed one‐factors is as large as possible. Constructions both for dihedral groups and for some classes of abelian groups are given. © 2002 John Wiley & Sons, Inc. J Combin Designs 10: 1–16, 2002