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Expansion in finite simple groups of Lie type

Emmanuel Breuillard, Ben J. Green, Robert Guralnick, Terence Tao (2015)

Journal of the European Mathematical Society

We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper [BGGT].

Extending the structural homomorphism of LCC loops

Piroska Csörgö (2005)

Commentationes Mathematicae Universitatis Carolinae

A loop Q is said to be left conjugacy closed if the set A = { L x / x Q } is closed under conjugation. Let Q be an LCC loop, let and be the left and right multiplication groups of Q respectively, and let I ( Q ) be its inner mapping group, M ( Q ) its multiplication group. By Drápal’s theorem [3, Theorem 2.8] there exists a homomorphism Λ : I ( Q ) determined by L x R x - 1 L x . In this short note we examine different possible extensions of this Λ and the uniqueness of these extensions.

Extensive varieties

Jaroslav Ježek, Tomáš Kepka (1975)

Acta Universitatis Carolinae. Mathematica et Physica

Finite simple zeropotent paramedial groupoids

Jung R. Cho, Tomáš Kepka (2002)

Czechoslovak Mathematical Journal

The study of paramedial groupoids (with emphasis on the structure of simple paramedial groupoids) was initiated in [1] and continued in [2], [3] and [5]. The aim of the present paper is to give a full description of finite simple zeropotent paramedial groupoids (i.e., of finite simple paramedial groupoids of type (II)—see [2]). A reader is referred to [1], [2], [3] and [7] for notation and various prerequisites.

F-quasigroups and generalized modules

Tomáš Kepka, Michael K. Kinyon, Jon D. Phillips (2008)

Commentationes Mathematicae Universitatis Carolinae

In Kepka T., Kinyon M.K., Phillips J.D., The structure of F-quasigroups, J. Algebra 317 (2007), 435–461, we showed that every F-quasigroup is linear over a special kind of Moufang loop called an NK-loop. Here we extend this relationship by showing an equivalence between the class of (pointed) F-quasigroups and the class corresponding to a certain notion of generalized module (with noncommutative, nonassociative addition) for an associative ring.

F-quasigroups isotopic to groups

Tomáš Kepka, Michael K. Kinyon, Jon D. Phillips (2010)

Commentationes Mathematicae Universitatis Carolinae

In Kepka T., Kinyon M.K., Phillips J.D., The structure of F-quasigroups, math.GR/0510298, we showed that every loop isotopic to an F-quasigroup is a Moufang loop. Here we characterize, via two simple identities, the class of F-quasigroups which are isotopic to groups. We call these quasigroups FG-quasigroups. We show that FG-quasigroups are linear over groups. We then use this fact to describe their structure. This gives us, for instance, a complete description of the simple FG-quasigroups. Finally,...

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