Omega-1 of maximal among Abelian normal subgroups with maximum rank in odd-order p-group equals omega-1 of centralizer
From Groupprops
This article states and (possibly) proves a fact that is true for odd-order p-groups: groups of prime power order where the underlying prime is odd. The statement is false, in general, for groups whose order is a power of two.
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Statement
Suppose is an odd prime, and is an group of prime power order where the prime is . Suppose is Maximal among Abelian normal subgroups (?) of , such that is also an Abelian normal subgroup of maximum rank (?) in . Then:
.
Here, denotes the first omega subgroup and denotes the centralizer inside .
Related facts
References
Journal references
- Solvability of groups of odd order by Walter Feit and John Griggs Thompson, Pacific Journal of Mathematics, Volume 13, Page 775 - 1029(Year 1963): This 255-page long paper gives a proof that odd-order implies solvable: any odd-order group (i.e., any finite group whose order is odd) is a solvable group.^{Project Euclid page}^{More info}, Page 796, Lemma 8.3, Section 8 (Miscellaneous Preliminary Lemmas), Chapter 2.
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 201, Lemma 4.14, Section 5.4, p-groups of small depth, ^{More info}