1,516 research outputs found
Maximal subgroups of direct products
We determine all maximal subgroups of the direct product \sc G^n of \sc n
copies of a group~\sc G. If \sc G is finite, we show that the number of
maximal subgroups of~\sc G^n is a quadratic function of~\sc n if \sc G is
perfect, but grows exponentially otherwise. We~deduce a theorem of Wiegold
about the growth behaviour of the number of generators of~\sc G^n.Comment: Plain TeX file, 8 page
The primitive idempotents of the p-permutation ring
Let G be a finite group, let p be a prime number, and let K be a field of
characteristic 0 and k be a field of characteristic p, both large enough. In
this note we state explicit formulae for the primitive idempotents of K\otimes
pp_k(G), where pp_k(G) is the ring of p-permutation kG-modules
Lifting endo--permutation modules
We prove that all endo--permutation modules for a finite group are
liftable from characteristic to characteristic
The algebra of Boolean matrices, correspondence functors, and simplicity
We determine the dimension of every simple module for the algebra of the
monoid of all relations on a finite set (i.e. Boolean matrices). This is in
fact the same question as the determination of the dimension of every
evaluation of a simple correspondence functor. The method uses the theory of
such functors developed in [BT2, BT3], as well as some new ingredients in the
theory of finite lattices.Comment: arXiv admin note: text overlap with arXiv:1510.0303
The algebra of essential relations on a finite set
Let X be a finite set and let k be a commutative ring. We consider the
k-algebra of the monoid of all relations on X, modulo the ideal generated by
the relations factorizing through a set of cardinality strictly smaller than
Card(X), called inessential relations. This quotient is called the essential
algebra associated to X. We then define a suitable nilpotent ideal of the
essential algebra and describe completely the structure of the corresponding
quotient, a product of matrix algebras over suitable group algebras. In
particular, we obtain a description of the Jacobson radical and of all the
simple modules for the essential algebra
The torsion group of endotrivial modules
Let G be a finite group and let T(G) be the abelian group of equivalence
classes of endotrivial kG-modules, where k is an algebraically closed field of
characteristic p. We determine, in terms of the structure of G, the kernel of
the restriction map from T(G) to T(S), where S is a Sylow p-subgroup of G, in
the case when S is abelian. This provides a classification of all torsion
endotrivial kG-modules in that case
Simple biset functors and double Burnside rings
Let G be a finite group and let k be a field. Our purpose is to investigate
the simple modules for the double Burnside ring kB(G,G). It turns out that they
are evaluations at G of simple biset functors. For a fixed finite group H, we
introduce a suitable bilinear form on kB(G,H) and we prove that the quotient of
kB(-,H) by the radical of the bilinear form is a semi-simple functor. This
allows for a description of the evaluation of simple functors, hence of simple
modules for the double Burnside ring
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