Employment of Minimal Generating Sets and Structure of Sylow 2-Subgroups Alternating Groups in Block Ciphers

Abstract

In this article, the research of Sylow p-subgroups of ${{A}_{n}}$ and ${{S}_{n}}$, which was started in Dmitruk and Suschansky (Structure of 2-Sylow subgroup of symmetric and alternating group, UMJ, N. 3, pp. 304–312, 1981, [1]), Skuratovskii (Cybern Syst Anal (1):27–41, 2009, [2]), Pawlik (Algebr Discret Math 21(2):264–281, 2016, [3]) is continued. Let $syl_2{A_{2^k}}$ and $syl_2{A_{n}}$ be Sylow 2-subgroups of the corresponding alternating groups $A_{2^k}$ and $A_{n}$. We find a minimal generating set and the structure for such subgroups $sy{{l}_{2}}{{A}_{{{2}^{k}}}}$ and $syl_2{A_{n}}$. The purpose of this paper is to research the structure of Sylow 2-subgroups and to construct a minimal generating set for such subgroups. The main result is to prove minimality of this generating set for the above indicated subgroups and also to describe their structure.