On the second commutants of finite Alperin groups

We refer to an Alperin group as a group in which the commutant of every 2-generated subgroup is cyclic. Alperin proved that if p is an odd prime then all finite p-groups with the property are metabelian. Nevertheless, finite Alperin 2-groups may fail to be metabelian. We prove that for each finite abelian group H there exists a finite Alperin group G for which G″ is isomorphic to H.     

Ghosts in modular representation theory

A ghost over a finite p-group G is a map between modular representations of G which is invisible in Tate cohomology. Motivated by the failure of the generating hypothesis—the statement that ghosts between finite-dimensional G-representations factor through a projective—we define the ghost number of kG to be the smallest integer l such that the composite of any l ghosts between finite-dimensional G-representations factors through a projective. In this paper we study ghosts and the ghost numbers of p-groups. We begin by showing that a weaker version of the generating hypothesis, where the target of the ghost is fixed to be the trivial representation k, holds for all p-groups. We then compute the ghost numbers of all cyclic p-groups and all abelian 2-groups with C₂ as a summand. We obtain bounds on the ghost numbers for abelian p-groups and for all 2-groups which have a cyclic subgroup of index 2. Using these bounds we determine the finite abelian groups which have ghost number at most 2. Our methods involve techniques from group theory, representation theory, triangulated category theory, and constructions motivated from homotopy theory.     

Transfer and Tate’s theorem

Let H be a subgroup of a finite group G, and assume that p is a prime that does not divide |G : H|. In favorable circumstances, one can use transfer theory to deduce that the largest abelian p-groups that occur as factor groups of G and of H are isomorphic. When this happens, Tate’s theorem guarantees that the largest not-necessarily-abelian p-groups that occur as factor groups of G and H are isomorphic. Known proofs of Tate’s theorem involve cohomology or character theory, but in this paper, a new elementary proof is given. It is also shown that the largest abelian p-factor group of G is always isomorphic to a direct factor of the largest abelian p-factor group of H. Received: 17 June 2008     

On finite Alperin 2-groups with elementary abelian second commutants

By an Alperin group we mean a group in which the commutant of each 2-generated subgroup is cyclic. Alperin proved that if p is an odd prime then all finite p-groups with this property are metabelian. The today??s actual problem is the construction of examples of nonmetabelian finite Alperin 2-groups. Note that the author had given some examples of finite Alperin 2-groups with second commutants isomorphic to Z ₂ and Z ₄ and proved the existence of finite Alperin 2-groups with cyclic second commutants of however large order by appropriate examples. In this article the existence is proved of finite Alperin 2-groups with abelian second commutants of however large rank.     

Connected components of the category of elementary abelian p-subgroups

We determine the maximal number of conjugacy classes of maximal elementary abelian subgroups of rank 2 in a finite p-group G, for an odd prime p. Namely, it is p if G has rank at least 3 and it is p+1 if G has rank 2. More precisely, if G has rank 2, there are exactly 1,2,p+1, or possibly 3 classes for some 3-groups of maximal nilpotency class.     

On the cohomology of highly connected covers of finite Hopf spaces

Relying on the computation of the André-Quillen homology groups for unstable Hopf algebras, we prove that if the mod p cohomology of both the fiber and the base in an H-fibration is finitely generated as algebra over the Steenrod algebra, then so is the mod p cohomology of the total space. In particular, the mod p cohomology of the n-connected cover of a finite H-space is always finitely generated as algebra over the Steenrod algebra.     

Finite p-Groups All of Whose Minimal Nonnormal Subgroups Posses Large Normal Closures

We obtained some results about finite p-groups G with G/H^G being abelian for all nonnormal subgroups H, where H^G denotes the normal closure of H. Moreover, we give a classification of finite p-groups G with G/H^G being cyclic for all nonnormal subgroups H.     

Maximal abelian and minimal nonabelian subgroups of some finite two-generator p-groups especially metacyclic

We study the subgroup structure of some two-generator p-groups and apply the obtained results to metacyclic p-groups. For metacyclic p-groups G, p > 2, we do the following: (a) compute the number of nonabelian subgroups with given derived subgroup, show that (ii) minimal nonabelian subgroups have equal order, (c) maximal abelian subgroups have equal order, (d) every maximal abelian subgroup is contained in a minimal nonabelian subgroup and all maximal subgroups of any minimal nonabelian subgroup are maximal abelian in G. We prove the same results for metacyclic 2-groups (e) with abelian subgroup of index p, (f) without epimorphic image ? D₈. The metacyclic p-groups containing (g) a minimal nonabelian subgroup of order p ⁴, (h) a maximal abelian subgroup of order p ³ are classified. We also classify the metacyclic p-groups, p > 2, all of whose minimal nonabelian subgroups have equal exponent. It appears that, with few exceptions, a metacyclic p-group has a chief series all of whose members are characteristic.     

Primitives and central detection numbers in group cohomology

Fix a prime p. Given a finite group G, let H^∗(G) denote its mod p cohomology. In the early 1990s, Henn, Lannes, and Schwartz introduced two invariants d₀(G) and d₁(G) of H^∗(G) viewed as a module over the mod p Steenrod algebra. They showed that, in a precise sense, H^∗(G) is respectively detected and determined by Hd(CG(V)) for d?d₀(G) and d?d₁(G), with V running through the elementary abelian p-subgroups of G.The main goal of this paper is to study how to calculate these invariants. We find that a critical role is played by the image of the restriction of H^∗(G) to H^∗(C), where C is the maximal central elementary abelian p-subgroup of G. A measure of this is the top degree e(G) of the finite dimensional Hopf algebra H^∗(C)_⊗H^∗(G)Fp, a number that tends to be quite easy to calculate.Our results are complete when G has a p-Sylow subgroup P in which every element of order p is central. Using the Benson-Carlson duality, we show that in this case, d₀(G)=d₀(P)=e(P), and a similar exact formula holds for d₁. As a bonus, we learn that He(G)(P) contains nontrivial essential cohomology, reproving and sharpening a theorem of Adem and Karagueuzian.In general, we are able to show that d₀(G)?max{e(CG(V))|V<G} if certain cases of Benson's Regularity Conjecture hold. In particular, this inequality holds for all groups such that the difference between the p-rank of G and the depth of H^∗(G) is at most 2. When we look at examples with p=2, we learn that d₀(G)?14 for all groups with 2-Sylow subgroup of order up to 64, with equality realized when G=SU(3,4).En route we study two objects of independent interest. If C is any central elementary abelian p-subgroup of G, then H^∗(G) is an H^∗(C)-comodule, and we prove that the subalgebra of H^∗(C)-primitives is always Noetherian of Krull dimension equal to the p-rank of G minus the p-rank of C. If the depth of H^∗(G) equals the rank of Z(G), we show that the depth essential cohomology of G is nonzero (reproving and extending a theorem of Green), and Cohen-Macauley in a certain sense, and prove related structural results.     

Locally Finite Groups with All Subgroups Normal-by-(Finite Rank)

A group is said to have finite (special) rank ≤ sif all of its finitely generated subgroups can be generated byselements. LetGbe a locally finite group and suppose thatH/H_Ghas finite rank for all subgroupsHofG, whereH_Gdenotes the normal core ofHinG. We prove that thenGhas an abelian normal subgroup whose quotient is of finite rank (Theorem 5). If, in addition, there is a finite numberrbounding all of the ranks ofH/H_G, thenGhas an abelian subgroup whose quotient is of finite rank bounded in terms ofronly (Theorem 4). These results are based on analogous theorems on locally finitep-groups, in which case the groupGis also abelian-by-finite (Theorems 2 and 3).     

Classifying spaces of Kač-Moody groups

We study the structure of classifying spaces of Kač-Moody groups from a homotopy theoretic point of view. They behave in many respects as in the compact Lie group case. The mod p cohomology algebra is noetherian and Lannes'T functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p-subgroups. Also, spaces of maps from classifying spaces of finite p-groups to classifying spaces of Kač-Moody groups are described in terms of classifying spaces of centralizers while the classifying space of a Kač-Moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p-subgroups, up to p-completion. We show that these properties are common to a larger class of groups, also including parabolic subgroups of Kač-Moody groups, and centralizers of finite p-subgroups. Received: 15 June 2000 / in final form: 20 September 2001 / Published online: 29 April 2002     

Minimal Number of Generators and Minimum Order of a Non-Abelian Group Whose Elements Commute with Their Endomorphic Images

A group in which every element commutes with its endomorphic images is called an “E-group″. If p is a prime number, a p-group G which is an E-group is called a “pE-group″. Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p ⁸ for any odd prime number p and this order is 2⁷ for p = 2. Some of these results are proved for a class wider than the class of E-groups.     

On maximal abelian subgroups in finite 2-groups

We determine here up to isomorphism the structure of any finite nonabelian 2-group G in which every two distinct maximal abelian subgroups have cyclic intersection. We obtain five infinite classes of such 2-groups (Theorem 1.1). This solves for p = 2 the problem Nr. 521 stated by Berkovich (in preparation). The more general problem Nr. 258 stated by Berkovich (in preparation) about the structure of finite nonabelian p-groups G such that A ∩ B = Z(G) for every two distinct maximal abelian subgroups A and B is treated in Theorems 3.1 and 3.2. In Corollary 3.3 we get a new result for an arbitrary finite 2-group. As an application of Theorems 3.1 and 3.2, we solve for p = 2 a problem of Heineken-Mann (Problem Nr. 169 stated in Berkovich, in preparation), classifying finite 2-groups G such that A/Z(G) is cyclic for each maximal abelian subgroup A (Theorem 4.1).      

子群的核平凡或正规闭包极大的有限p群

称有限 p 群 G 为ACT 群,如果对每个交换子群H, 其正规核 H_G=1 或 H_G=H. 又称p 群 G是CC 群,如果对每个非正规交换子群H, 有 H_G=1 或 H^G 在G中的指数为 p. 本文分类了ACT 群和CC 群.     

Character tables of p-groups with derived subgroup of prime order I

Two character tables of finite groups are isomorphic if there exist a bijection for the irreducible characters and a bijection for the conjugacy classes that preserve all the character values. We give necessary and sufficient conditions for two finite groups to have isomorphic character tables. In the case of finite p-groups with derived subgroup of order p, we show that the character tables can be classified by equivalence classes of certain homomorphisms of abelian p-groups.     

On equality of central and class preserving automorphisms of finite p-groups

Let G be a finite non-abelian p-group, where p is a prime. Let Aut_c(G) and Aut_z(G) respectively denote the group of all class preserving and central automorphisms of G. We give a necessary and sufficient condition for G such that Aut_c(G) = Aut_z(G) and classify all finite non-abelian p-groups G with elementary abelian or cyclic center such that Aut_c(G) = Aut_z(G). We also characterize all finite p-groups G of order ≤ p ⁷ such that Aut_z(G) = Aut_z(G) and complete the classification of all finite p-groups of order ≤ p ⁵ for which there exist non-inner class preserving automorphisms.     

Group algebra series and coboundary modules

The shift action on the 2-cocycle group Z²(G,C) of a finite group G with coefficients in a finitely generated abelian group C has several useful applications in combinatorics and digital communications, arising from the invariance of a uniform distribution property of cocycles under the action. In this article, we study the shift orbit structure of the coboundary subgroup B²(G,C) of Z²(G,C). The study is placed within a well-known setting involving the Loewy and socle series of a group algebra over G. We prove new bounds on the dimensions of terms in such series. Asymptotic results on the size of shift orbits are also derived; for example, if C is an elementary abelian p-group, then almost all shift orbits in B²(G,C) are maximal-sized for large enough finite p-groups G of certain classes.     

Essential cohomology of finite groups

In this paper we prove that a finite group G with Cohen-Macaulay mod p cohomology will have non-trivial undetectable elements in if and only if G is a p-group such that every element of order p in G is central. Applications and examples are also provided. Received: April 18, 1996     

Embeddings of minimal non-abelian p-groups

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. For a given finite group G, let p(G) denote the minimal degree of a faithful representation of G by permutation matrices, and let c(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices. See [4]. It is easy to see that c(G) is a lower bound for p(G). Behravesh [H. Behravesh, The minimal degree of a faithful quasi-permutation representation of an abelian group, Glasg. Math. J. 39 (1) (1997) 51-57] determined c(G) for every finite abelian group G and also [H. Behravesh, Quasi-permutation representations of p-groups of class 2, J. Lond. Math. Soc. (2) 55 (2) (1997) 251-260] gave the algorithm of c(G) for each finite group G. In this paper, we first improve this algorithm and then determine c(G) and p(G) for an arbitrary minimal non-abelian p-group G.