Minimal Systems of Generators and Relations and Properties of Wreath Products of Perfect Groups
DOI:
https://doi.org/10.20535/1810-0546.2014.4.28406Keywords:
Minimal system of generators, Wreath product, Metaperfectgroup, GroupAbstract
Generators and defining relations for wreath products of perfect group which is two generating and alternating groups \[A_{n_1}\imath A_{n_2}\imath ...\imath A_{n_m},\] (m > 2 times) are given. System of generators of metaperfect groups are found. Generators and defining relations for wreath products of 2-generating perfect groups were found, including alternating groups, i.e. \[A_{n_1}\imath A_{n_2}\imath ...\imath A_{n_m},\] (m > 2 time). Systems of generators for metaperfect groups were investigated. A constructive proof of the minimality found system of generators was presented. It is shown that metaperfect group is not locally finite group. Cases of wreath product \[G\imath \mathit{D}\] of metaperfect group D with group (G, X), which may be such that acts as a transitive and intransitive, construct the corresponding systems of generators. Presented generalization is the appearance of the product of different perfect groups \[H_{i}\] and finding the exact value \[d_{n}^{wr}(H_{i})\] instead of the estimate. As it was found, for perfect 2-generator groups, wich has conditions founded by us, satisfy the equality of lower estimate \[d_{n}^{wr}(H_{i})=d(H_{i})=2,\] it was easily generalized for 3-generated groups as \[d_{n}^{wr}(H_{i})=3.\]. It was found, that some of the properties immanent alternating groups are saved for metaalternating groups. The criterion of perfectly for metaalternating group was obtained. Inverse limit of metaalternating group, which іs proved to be branch group that does not own the property of locally finiteness was analyzed.
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