| Summary: | The structure of the commutator subgroup of Sylow 2-subgroups of an alternating group <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <msup> <mn>2</mn> <mi>k</mi> </msup> </msub> </semantics> </math> </inline-formula> is determined. This work continues the previous investigations of me, where minimal generating sets for Sylow 2-subgroups of alternating groups were constructed. Here we study the commutator subgroup of these groups. The minimal generating set of the commutator subgroup of <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <msup> <mn>2</mn> <mi>k</mi> </msup> </msub> </semantics> </math> </inline-formula> is constructed. It is shown that <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mrow> <mo>(</mo> <mi>S</mi> <mi>y</mi> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>A</mi> <msup> <mn>2</mn> <mi>k</mi> </msup> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mi>S</mi> <mi>y</mi> <msubsup> <mi>l</mi> <mn>2</mn> <mo>′</mo> </msubsup> <msub> <mi>A</mi> <msup> <mn>2</mn> <mi>k</mi> </msup> </msub> <mo>,</mo> <mspace width="0.166667em"></mspace> <mi>k</mi> <mo>></mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>. It serves to solve quadratic equations in this group, as were solved by Lysenok I. in the Grigorchuk group. It is proved that the commutator length of an arbitrary element of the iterated wreath product of cyclic groups <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>C</mi> <msub> <mi>p</mi> <mi>i</mi> </msub> </msub> <mo>,</mo> <mspace width="0.166667em"></mspace> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </semantics> </math> </inline-formula> equals to 1. The commutator width of direct limit of wreath product of cyclic groups is found. Upper bounds for the commutator width <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>c</mi> <mi>w</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> of a wreath product of groups are presented in this paper. A presentation in form of wreath recursion of Sylow 2-subgroups <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>y</mi> <msub> <mi>l</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>A</mi> <msup> <mn>2</mn> <mi>k</mi> </msup> </msub> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> of <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <msup> <mn>2</mn> <mi>k</mi> </msup> </msub> </semantics> </math> </inline-formula> is introduced. As a result, a short proof that the commutator width is equal to 1 for Sylow 2-subgroups of alternating group <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <msup> <mn>2</mn> <mi>k</mi> </msup> </msub> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>></mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>, the permutation group <inline-formula> <math display="inline"> <semantics> <msub> <mi>S</mi> <msup> <mn>2</mn> <mi>k</mi> </msup> </msub> </semantics> </math> </inline-formula>, as well as Sylow <i>p</i>-subgroups of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>y</mi> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>A</mi> <msup> <mi>p</mi> <mi>k</mi> </msup> </msub> </mrow> </semantics> </math> </inline-formula> as well as <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>y</mi> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>S</mi> <msup> <mi>p</mi> <mi>k</mi> </msup> </msub> </mrow> </semantics> </math> </inline-formula>) are equal to 1 was obtained. A commutator width of permutational wreath product <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mo>≀</mo> <msub> <mi>C</mi> <mi>n</mi> </msub> </mrow> </semantics> </math> </inline-formula> is investigated. An upper bound of the commutator width of permutational wreath product <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mo>≀</mo> <msub> <mi>C</mi> <mi>n</mi> </msub> </mrow> </semantics> </math> </inline-formula> for an arbitrary group <i>B</i> is found. The size of a minimal generating set for the commutator subgroup of Sylow 2-subgroup of the alternating group is found. The proofs were assisted by the computer algebra system GAP.
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