A p-local finite group consists of a finite p-group S, together with a pair of categories which encode “conjugacy” relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we exa...

Let $H$, $L$ and $X$ be subgroups of a finite group$G$. Then $H$ is said to be $X$-permutable with $L$ if for some$xin X$ we have $AL^{x}=L^{x}A$. We say that $H$ is emph{$X$-quasipermutable } (emph{$X_{S}$-quasipermutable}, respectively) in $G$ provided $G$ has a subgroup$B$ such that $G=N_{G}(H)B$ and $H$ $X$-permutes with $B$ and with all subgroups (with all Sylowsubgroups, respectively) $...

The Chermak–Delgado lattice of a finite group G is self-dual sublattice the subgroup G. In this paper, we focus on groups whose an elementary abelian p-group. We prove that such are nilpotent class 2. also that, for any p-group E, there exists E.

where D is some small category, F is a functor from D to the category of spaces, and, for each object d of D, F (d) has the homotopy type of BH for some subgroup H of G. An expression like 1.1 is sometimes called a homology approximation to BG or a homology decomposition of BG, and can be used either to make calculations with BG or to prove general theorems about BG by induction. (Of course an ...

For a finite p-group P the following three conditions are equivalent: (a) to have a (proper) partition, that is, to be the union of some proper subgroups with trivial pairwise intersections; (b) to have a proper subgroup outside of which all elements have order p; (c) to be a semidirect product P = P1o ⟨φ⟩ where P1 is a subgroup of index p and φ is a splitting automorphism of order p of P1. It ...

We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.

In this paper we consider the category C(̃k, H̃) of the (̃k, H̃)-modules, including all the Verma modules, where k is some compact Lie algebra and H some Cartan subgroup, k̃ and H̃ are the corresponding affine Lie algebra and the affine Cartan group, respectively. To this category we apply the Zuckermann functor and its derivatives. By using the determinant bundle structure, we prove the natural dual...