View source for Omega-1 of maximal among Abelian normal subgroups with maximum rank in odd-order p-group equals omega-1 of centralizer

{{odd-order-only p-group statement}}

==Statement==

Suppose <math>p</math> is an odd prime, and <math>P</math> is an [[group of prime power order]] where the prime is <math>p</math>. Suppose <math>A</math> is [[fact about::maximal among Abelian normal subgroups]] of <math>P</math>, such that <math>A</math> is ''also'' an [[fact about::Abelian normal subgroup of maximum rank]] in <math>P</math>. Then:

<math>\Omega_1(C_P(\Omega_1(A))) = \Omega_1(A)</math>.

Here, <math>\Omega_1</math> denotes the [[fact about::omega subgroups of a group of prime power order|first omega subgroup]] and <math>C_P</math> denotes the [[centralizer]] inside <math>P</math>.

==Related facts==

* [[Abelian normal subgroup of maximum rank that is also maximal among Abelian normal subgroups exists]]

==References==

===Journal references===

* {{paperlink|FeitThompson}}, Page 796, Lemma 8.3, Section 8 (''Miscellaneous Preliminary Lemmas''), Chapter 2.
===Textbook references===

* {{booklink-proved|Gorenstein|201|Lemma 4.14, Section 5.4, ''p-groups of small depth''}}