Abelian-to-normal replacement theorem for prime-cube index for odd prime
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This article defines a replacement theorem
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Contents
History
The result appears to have been first proved in a paper by Alperin in 1965 and later proved as part of a larger array of results in a paper by David Jonah and Marc Konvisser in 1975.
Statement
Suppose is an odd prime. Then, if a finite
-group contains an abelian subgroup of index
, it contains an abelian normal subgroup (hence, an Abelian normal subgroup of group of prime power order (?)) of index
.
Facts used
- Jonah-Konvisser congruence condition on number of abelian subgroups of prime-square index for odd prime
- Prime power order implies nilpotent, nilpotent implies every maximal subgroup is normal
Proof
The Jonah-Konvisser proof
Given: An odd prime , a finite
-group
, an abelian subgroup
of
of index
.
To prove: has an abelian normal subgroup of index
.
Proof:
- Let
be a maximal subgroup of
containing
: Such a
exists and has index
in
.
- By fact (1), the number of abelian subgroups of
of index
is either equal to
or congruent to
modulo
.
- If
has exactly two abelian subgroups of index
, then both of them are normal in
: Since
is maximal in
, it is normal (See fact (2)). Thus, any inner automorphism of
sends subgroups of
to subgroups of
. Since the sizes of orbits are either
or powers of
, and there are only two abelian subgroups of index
, they must both be normal in
.
- If the number of abelian subgroups of
of index
is congruent to
modulo
, then there exists a subgroup of index
in
that is normal in
: The inner automorphisms of
permute abelian subgroups of index
in
, since
is normal in
. All orbits have size either
or a nontrivial power (and hence, multiple) of
. Since the total number is
modulo
, there must be at least one orbit of size one.
Alperin's proof
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Journal references
- Large abelian subgroups of p-groups by Jonathan Lazare Alperin, Transactions of the American Mathematical Society, Volume 117, Page 10 - 20(Year 1965): Official copyMore info
- Counting abelian subgroups of p-groups: a projective approach by Marc Konvisser and David Jonah, Journal of Algebra, ISSN 00218693, Volume 34, Page 309 - 330(Year 1975): PDF (ScienceDirect)More info