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On central nilpotency in finite loops with nilpotent inner mapping groups

Markku Niemenmaa, Miikka Rytty (2008)

Commentationes Mathematicae Universitatis Carolinae

In this paper we consider finite loops whose inner mapping groups are nilpotent. We first consider the case where the inner mapping group I ( Q ) of a loop Q is the direct product of a dihedral group of order 8 and an abelian group. Our second result deals with the case where Q is a 2 -loop and I ( Q ) is a nilpotent group whose nonabelian Sylow subgroups satisfy a special condition. In both cases it turns out that Q is centrally nilpotent.

On centrally nilpotent loops

L. V. Safonova, K. K. Shchukin (2000)

Commentationes Mathematicae Universitatis Carolinae

Using a lemma on subnormal subgroups, the problem of nilpotency of multiplication groups and inner permutation groups of centrally nilpotent loops is discussed.

On certain homotopy actions of general linear groups on iterated products

Ran Levi, Stewart Priddy (2001)

Annales de l’institut Fourier

The n -fold product X n of an arbitrary space usually supports only the obvious permutation action of the symmetric group Σ n . However, if X is a p -complete, homotopy associative, homotopy commutative H -space one can define a homotopy action of GL n ( p ) on X n . In various cases, e.g. if multiplication by p r is null homotopic then we get a homotopy action of G L n ( / p r ) for some r . After one suspension this allows one to split X n using idempotents of 𝔽 p GL n ( / p ) which can be lifted to 𝔽 p GL n ( / p r ) . In fact all of this is possible if X is an H -space...

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