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

Author:
Kenneth Hickin

Journal:
Trans. Amer. Math. Soc. **290** (1985), 457-481

MSC:
Primary 20F99; Secondary 20E06, 20E15, 20E28, 20F28

MathSciNet review:
792807

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Abstract: In we extend the results of [ ] on centralizers to r.e. subgroups and show, e.g., that every a.c. group has an -equivalent subgroup of the same power which is embedded maximally in itself; and we pursue a natural typology of maximal subgroups. shows that if is a countable group of automorphisms of a countable a.c. group such that , then there exists such that the extension ( for all ) is a subgroup of . We show in 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 [ ]. Finally, in , also using Ziegler's construction we show that there exists an a.c. group of any power and having any countable skeleton which has a free subgroup such that for all and there exist free generators such that .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0792807-2

Keywords:
Algebraically closed groups,
automorphism groups,
maximal subgroups,
extensions

Article copyright:
© Copyright 1985
American Mathematical Society