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Nonsplitting F-quasigroups

Stephen Gagola III (2012)

Commentationes Mathematicae Universitatis Carolinae

T. Kepka, M.K. Kinyon and J.D. Phillips: The structure of F-quasigroups, J. Algebra 317 (2007), no. 2, 435–461 developed a connection between F-quasigroups and NK-loops. Since NK-loops are contained in the variety generated by groups and commutative Moufang loops, a question that arises is whether or not there exists a nonsplit NK-loop and likewise a nonsplit F-quasigroup. Here we prove that there do indeed exist nonsplit F-quasigroups and show that there are exactly four corresponding nonsplit...

Normal Subgroup of Product of Groups

Hiroyuki Okazaki, Kenichi Arai, Yasunari Shidama (2011)

Formalized Mathematics

In [6] it was formalized that the direct product of a family of groups gives a new group. In this article, we formalize that for all j ∈ I, the group G = Πi∈IGi has a normal subgroup isomorphic to Gj. Moreover, we show some relations between a family of groups and its direct product.

Normalizers and self-normalizing subgroups II

Boris Širola (2011)

Open Mathematics

Let 𝕂 be a field, G a reductive algebraic 𝕂 -group, and G 1 ≤ G a reductive subgroup. For G 1 ≤ G, the corresponding groups of 𝕂 -points, we study the normalizer N = N G(G 1). In particular, for a standard embedding of the odd orthogonal group G 1 = SO(m, 𝕂 ) in G = SL(m, 𝕂 ) we have N ≅ G 1 ⋊ µm( 𝕂 ), the semidirect product of G 1 by the group of m-th roots of unity in 𝕂 . The normalizers of the even orthogonal and symplectic subgroup of SL(2n, 𝕂 ) were computed in [Širola B., Normalizers and self-normalizing...

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