  In this article the investigation of Sylows p-subgroups of ${{A}_{n}}$ and
${{S}_{n}}$, which was started in article of U. Dmitruk, V. Suschansky
"Structure of 2-sylow subgroup of symmetric and alternating group" and article
of R.~Skuratovskii "Corepresentation of a Sylow p-subgroup of a group S_n"
\cite{Dm, Sk, Paw} is continued. Let $Syl_2{A_{2^k}}$ and $Syl_2{A_{n}}$ be
Sylow 2-subgroups of corresponding alternating groups $A_{2^k}$ and $A_{n}$. We
find a least generating set and a structure for such subgroups
$Sy{{l}_{2}}{{A}_{{{2}^{k}}}}$ and $Syl_2{A_{n}}$ and commutator width of
$Syl_2{A_{2^k}}$ \cite{Mur}.
  The authors of \cite{Dm, Paw} 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 purpose of this paper is to research the structure of a 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 the Sylow 2-subgroup of alternating group of degree $2^k $
and proof its minimality.
  The main result is the proof of minimality of this generating set of the
above described subgroups and also the description of their structure.
