A characterization of endotrivial modules over p-groups

Let P be a finite p-group, and let k a field of characteristic p>0. We prove that a finietly generated kG-module M is endotrivial if and only if

A class of special p-groups

Starting by the multiplication table of a finite group G, for each odd prime p a special p-group P of exponent p is constructed. Some connections between the structures of G and P, concerning subgroups and automorphisms, are given. When p does not divide the order of G, for every normal subgroup of G a direct decomposition of P is done. Besides, the order of the derived subgroup of G is proved to be connected with the existence of some abelian subgroups of maximal order in P. Finally, the elements of P with large centralizers are characterized in terms of algebraic properties of the multiplication table of G. Dedicated to Herman Heineken on the occasion of his 60th birthday Work supported by M.U.R.S.T. and G.N.S.A.G.A-C.N.R. of Italy.     

Frattinian p-groups

The notion of a Frattinian p-group generalizes that of an extra-special p-group. We prove a central decomposition theorem and describe some relations to Frattini extensions and to automorphisms of finite p-groups.To Helmut Salzmann on his 60th birthday     

有限超特殊p-群的一个注记

本文获得了以下结果：设G为有限超特殊P-群．则下列条件等价：（1）G的非平凡特征子群的阶相同；（2）G的非平凡特征子群唯一；（3）当p〉2时，exp G=p；当P=2时，G≠D8．     

Disposition p-groups

For any prime p and positive integers c, d there is up to isomorphism a unique p-group \({G_{d}^{c}(p)}\) of least order having any (finite) p-group G with rank \({d(G) \le d}\) and Frattini class \({c_{p}(G) \le c}\) as epimorphic image. Here \({c_{p}(G) = n}\) is the least positive integer such that G has a central series of length n with all factors being elementary. This “disposition” p-group \({G_{d}^{c}(p)}\) has been examined quite intensively in the literature, sometimes controversially. The objective of this paper is to present a summary of the known facts, and to add some new results. For instance we show that for \({G = G_{d}^{c}(p)}\) the centralizer \({C_{G}(x) = \langle Z(G), x \rangle}\) whenever \({x \in G}\) is outside the Frattini subgroup, and that for odd p and \({d \ge 2}\) the group \({E = G_{d}^{c+1}(p)/(G_{d}^{c+1}(p))^{p^{c}}}\) is a distinguished Schur cover of G with \({E/Z(E) \cong G}\). We also have a fibre product construction of \({G_{d}^{c+1}(p)}\) in terms of \({G = G_{d}^{c}(p)}\) which might be of interest for Galois theory.     

On reduction of p-groups

We define the notion of irreducibility of a pgroup and show how any pgroup G can be reduced to an irreducible group H. We show that G is realizable as the Galois group of a regular extension of Q(T) if H is. Finally, we give some sufficient conditions on the

number of generators of a pgroup and the structure of its Frattini subgroup for it to be reducible to the trivial group.     

Transfers of metabelian p-groups

Explicit expressions for the transfers V _i from a metabelian p-group G of coclass cc(G)?=?1 to its maximal normal subgroups M ₁, . . . , M _p+1 are derived by means of relations for generators. The expressions for the exceptional case p?=?2 differ significantly from the standard case of odd primes p?≥ 3. In both cases the transfer kernels Ker(V _i ) are calculated and the principalisation type of the metabelian p-group is determined, if G is realised as the Galois group ${{\rm{Gal}}({F}_p^2(K)\vert K)}$ of the second Hilbert p-class field ${{F}_p^2(K)}$ of an algebraic number field K. For certain metabelian 3-groups G with abelianisation G/G′ of type (3, 3) and of coclass cc(G)?=?r?≥ 3, it is shown that the principalisation type determines the position of G on the coclass graph ${\mathcal{G}(3,r)}$ in the sense of Eick and Leedham-Green.     

A classification of some regular p-groups and its applications

In this paper we classify regular p-groups with type invariants (e, 1,1,1) for e≥2 and (1,1,1,1,1). As a by-product, we give a new approach to the classification of groups of order p5, p ≥ 5 a prime.     

交换子群较小的一类有限p群

研究了阶为p(m(m+1)/2)且交换子群的最大阶为p(m)的有限群,得到了这类特殊的p群的几个性质,给出了满足极大类条件的这类p群的同构分类.     

Note on elongations of totally projective p-groups by p ω+n -projective p-groups

It is proved that any Σ-group, which is a special elongation of a totally projective abelian p-group by a p ^ω+1-projective abelian p-group, is totally projective. In particular, each p ^ω+1-projective abelian Σ-p-group is a direct sum of countable p-groups of lengths not exceeding ω + 1. This strengthens our recent result published in Comment. Math. Univ. St. Pauli (2006). Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 180–185, April–June, 2006.     

Posets of subgroups in p-groups

It is shown that certain classes of finite p-groups are characterized by the ranked partially ordered set of conjugacy classes of their subgroups.     

奇数阶亚循环p-群分类的一个证明

一个有限p群p称为亚循环的,如果P有一个循环的正规子群A,使得PA是循环的.1973年King在文[2]中对亚循环p群进行了分类;在文[4,5]中M.F.NewmanandMingYaoXu发现了这些群的新的表示,给出了新的分类方法,这种方法是由p群生成算法(见文[3])得到的.本文的目的是给这些结果的另一种证明,与p群生成算法是不相关的.