a.c. groups: extensions, maximal subgroups, and automorphisms

Abstract:

In $\S 1$ we extend the results of [${\mathbf {3}}$] on centralizers to r.e. subgroups and show, e.g., that every a.c. group has an $\infty - \omega$-equivalent subgroup of the same power which is embedded maximally in itself; and we pursue a natural typology of maximal subgroups. $\S 2$ shows that if $A$ is a countable group of automorphisms of a countable a.c. group $G$ such that $A \supset \operatorname {Inn}\;G$, then there exists $\tau \in \operatorname {Aut}\;G$ such that the ${\text {HNN}}$ extension ($A,\tau :{\tau ^{ - 1}}g\tau = \tau (g)$ for all $g \in \operatorname {Inn}\;G$) is a subgroup of $\operatorname {Aut}\;G$. We show in $\S 3$ that every a.c. group with a countable skeleton has a proper extension to an a.c. group having any skeleton that contains the original one and any f.g. group which contains the countable a.c. group equivalent to the original one as an r.e. subset. This uses Ziegler’s construction [${\mathbf {7}}$]. Finally, in $\S 4$, also using Ziegler’s construction we show that there exists an a.c. group $A$ of any power and having any countable skeleton which has a free subgroup $M$ such that for all $x \in A - M$ and $y \in A$ there exist free generators $a,b,c \in M$ such that $y = {(ax)^b}{(ax)^c}$.