Structure, minimal generating systems and properties of Sylow 2-subgroups of alternating group

TL;DR

This paper investigates the structure of Sylow 2-subgroups of alternating groups, constructs minimal generating systems, and proves their minimality using automorphisms of binary tree portraits.

Contribution

It provides the first constructive proof of minimal generating sets for Sylow 2-subgroups of alternating groups and describes their detailed structure.

Findings

Constructed minimal generating systems for Sylow 2-subgroups.

Proved the minimality of these generating sets.

Described the detailed structure of these Sylow subgroups.

Abstract

The background of this paper is the following: search of the minimal systems of generators for this class of group which still was not founded also problem of representation for this class of group, exploration of systems of generators for Sylow 2-subgroups $Syl_2{A_{2^k}}$ and $Syl_2{A_{n}}$ of alternating group, finding structure of these subgroups. The authors of article "Structure of 2-sylow subgroup of symmetric and alternating group" U.~Dmitruk, V.~Suschansky didn't proof minimality of finding by them system of generators for such Sylow 2-subgroups of $A_n$ and structure of it were founded only descriptively. The aim of this paper is to research the structure of Sylow 2-subgroups and to construct a minimal generating system for such subgroups. In other words, the problem is not simply in the proof of existence of a generating set with elements for Sylow 2-subgroup of alternating group of degree $2^k$ and its constructive proof and proof its minimality. For the construction of minimal generating set we used the representation of elements of group by automorphisms of portraits for binary tree. Also, the goal of this paper is to investigate the structure of 2-sylow subgroup of alternating group more exactly and deep than in articlle of U.~Dmitruk and V.~Suschansky. The main result is the proof of minimality of this generating set of the above described subgroups and also the description of their structure. Key words: minimal system of generators; wreath product; Sylow subgroups; group; semidirect product.