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On the structure and zero divisors of the Cayley-Dickson sedenion algebra

Raoul E. Cawagas (2004)

Discussiones Mathematicae - General Algebra and Applications

The algebras ℂ (complex numbers), ℍ (quaternions), and 𝕆 (octonions) are real division algebras obtained from the real numbers ℝ by a doubling procedure called the Cayley-Dickson Process. By doubling ℝ (dim 1), we obtain ℂ (dim 2), then ℂ produces ℍ (dim 4), and ℍ yields 𝕆 (dim 8). The next doubling process applied to 𝕆 then yields an algebra 𝕊 (dim 16) called the sedenions. This study deals with the subalgebra structure of the sedenion algebra 𝕊 and its zero divisors. In particular, it shows...

On the structure of finite loop capable Abelian groups

Markku Niemenmaa (2007)

Commentationes Mathematicae Universitatis Carolinae

Loop capable groups are groups which are isomorphic to inner mapping groups of loops. In this paper we show that abelian groups C p k × C p × C p , where k ≥ 2 and p is an odd prime, are not loop capable groups. We also discuss generalizations of this result.

On the structure of finite loop capable nilpotent groups

Miikka Rytty (2010)

Commentationes Mathematicae Universitatis Carolinae

In this paper we consider finite loops and discuss the problem which nilpotent groups are isomorphic to the inner mapping group of a loop. We recall some earlier results and by using connected transversals we transform the problem into a group theoretical one. We will get some new answers as we show that a nilpotent group having either C p k × C p l , k > l ≥ 0 as the Sylow p -subgroup for some odd prime p or the group of quaternions as the Sylow 2 -subgroup may not be loop capable.

On the structure of finite paramedial quasigroups

V. A. Shcherbacov, D. I. Pushkashu (2010)

Commentationes Mathematicae Universitatis Carolinae

Information on the structure of finite paramedial quasigroups, including a classification of finite simple paramedial quasigroups, is given. The problem ``Classify the finite simple paramedial quasigroups'' was posed by J. Ježek and T. Kepka at the conference LOOPS'03, Prague 2003.

On the structure of groups whose non-abelian subgroups are subnormal

Leonid Kurdachenko, Sevgi Atlıhan, Nikolaj Semko (2014)

Open Mathematics

The main aim of this article is to examine infinite groups whose non-abelian subgroups are subnormal. In this sense we obtain here description of such locally finite groups and, as a consequence we show several results related to such groups.

On the structure of sequences with forbidden zero-sum subsequences

W. D. Gao, R. Thangadurai (2003)

Colloquium Mathematicae

We study the structure of longest sequences in ℤ ₙ d which have no zero-sum subsequence of length n (or less). We prove, among other results, that for n = 2 a and d arbitrary, or n = 3 a and d = 3, every sequence of c(n,d)(n-1) elements in ℤ ₙ d which has no zero-sum subsequence of length n consists of c(n,d) distinct elements each appearing n-1 times, where c ( 2 a , d ) = 2 d and c ( 3 a , 3 ) = 9 .

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