On w-maximal groups

Details On w-maximal groups On w-maximal groups

2010-03-31

arxiv.org/abs/1003.6048v1

Mathematics

Group Theory

15 pages

Scientific paper

Let $w = w(x_1,..., x_n)$ be a word, i.e. an element of the free group $F =$ on $n$ generators $x_1,..., x_n$. The verbal subgroup $w(G)$ of a group $G$ is the subgroup generated by the set $\{w (g_1,...,g_n)^{\pm 1} | g_i \in G, 1\leq i\leq n \}$ of all $w$-values in $G$. We say that a (finite) group $G$ is $w$-maximal if $|G:w(G)|> |H:w(H)|$ for all proper subgroups $H$ of $G$ and that $G$ is hereditarily $w$-maximal if every subgroup of $G$ is $w$-maximal. In this text we study $w$-maximal and hereditarily $w$-maximal (finite) groups.

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