有一个指数为$p^3$的循环子群的有限$p$群

We classify up to isomorphism those finite p-groups,for odd primes p,which contain a cyclic subgroup of index p3.     

Schur multiplicators of infinite pro-<Emphasis Type=Italic>p</Emphasis>-groups with finite coclass

Let G be an infinite pro-p-group of finite coclass and let M(G) be its Schur multiplicator. For p > 2, we determine the isomorphism type of Hom(M(G), ℤ_p), where ℤ_p denotes the p-adic integers, and show that M(G) is infinite. For p = 2, we investigate the Schur multiplicators of the infinite pro-2-groups of small coclass and show that M(G) can be infinite, finite or even trivial.     

Schur multiplicators of finite <Emphasis Type=Italic>p</Emphasis>-groups with fixed coclass

We investigate the Schur multiplicators M(G) of p-groups G using coclass theory. For p > 2 we show that there are at most finitely many p-groups G of coclass r with |M(G)| ≤ s for every r and s. We observe that this is not true for p = 2 by constructing infinite series of 2-groups G with coclass r and |M(G)| = 1. We investigate the Schur multiplicators of the 2-groups of coclass r further.     

Nonsolvable D 2-groups

Let G be a finite group. Let Irr₁(G) be the set of nonlinear irreducible characters of G and cd₁(G) the set of degrees of the characters in Irr₁(G). A group G is said to be a D₂-group if |cd₁(G)| = |Irr₁(G)| - 2. The main purpose of this paper is to classify nonsolvable D₂-groups.     

Generators of 2-groups

LetG be a finite 2-group. If each normal abelian subgroup ofG can be generated byd elements, then each subgroup ofG can be generated byd ²+1/2d(d+1) elements.     

Finite quaternion-free 2-groups

We present an elementary proof of the classification theorem for finite nonmodular quaternion-free 2-groups. This proof does not involve the structure theory of powerful 2-groups. Such a new proof is also necessary, since there are several gaps in the original proof given in [5].     

The Quillen categories of <Emphasis Type="Italic">p</Emphasis>-groups and coclass theory

Coclass theory can be used to define infinite families of finite p-groups of a fixed coclass. It is conjectured that the groups in one of these infinite families all have isomorphic mod-p cohomology rings. Here we prove that almost all groups in one of these infinite families have equivalent Quillen categories. We also show how the Quillen categories of the groups in an infinite family are connected to the Quillen category of their associated infinite pro-p-group of finite coclass.     

Finite separable metacyclic 2-groups

A finite separable metacyclic 2-group G can be written as the semidirect product of a cyclic group with another cyclic group. Necessary and sufficient conditions are given for when all other split decompositions of G result in, up to isomorphism, this same semidirect product representation for G.     

Minimal non-quaternion-free finite 2-groups

We classify minimal non-quaternion-free finite 2-groups which possess a nonmodular proper subgroup.     

Kummer's lemma for cyclic 2-groups

For cyclic 2-groupsC, we characterize the kernel of the map induced on the units of the integral group ring by the coefficient reduction . This allows us to prove that, for any finite abelian 2-groupA, the circular units of ℤA (i.e. those which are mapped to cyclotomic units by every character ofA) can be generated in a certain systematic way. Work supported in part by an NSERC (Canada) operating grant     

Order three normalizers of 2-groups

This paper examines order three elements of finite groups which normalize no nontrivial 2-subgroup. The motivation for finding such elements arises out of a problem in modular representation theory. The question of when these elements appear in the almost simple groups was posed by Robinson in the context of studying 2-blocks of defect zero. For the almost simple groups, a complete classification of order three elements with this property is determined. On the basis of this result, necessary conditions are then given for the existence of such elements in a large class of finite groups.