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The Frattini module and p'-automorphisms of free pro-p groups.
Communications in Arithmetic Fundamental Groups, Institute of Mathematical
Science Analysis 1267 (2002), Kyoto University, Research Institute for
Mathematical Sciences, Kyoto, 2002, 177-188.
Darren Semmen
Website: http://math.uci.edu/~dsemmen
Institution: University of California, Irvine
Abstract:
If a non-trivial subgroup A of the group of continuous
automorphisms
of a non-cyclic free pro-p group F has finite order, not
divisible by
p, then the group of fixed points
FixF{A}
has infinite rank.
The semi-direct product Fxs
A is the universal
p-Frattini cover of a finite group G, and so is the projective
limit
of a sequence of finite groups starting with G, each a canonical
group
extension of its predecessor by the Frattini module.
Examining appearances of the trivial simple module 1
in the Frattini module's Jordan-Hölder series arose in
investigations ([FK97], [BaFr02] and
[Sem02]) of modular towers. The number of these
appearances prevents FixF{A}
from having finite rank.