Elementary abelian-to-2-subnormal replacement theorem
From Groupprops
This article defines a replacement theorem
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Contents
Statement
Hands-on statement
Suppose is a Group of prime power order (?), say , where . Let be an elementary abelian subgroup of .
Then, there exists an elementary abelian subgroup of such that:
- has the same order as
- is a 2-subnormal subgroup of : it is normal in its normal closure in .
Statement in terms of weak 2-subnormal replacement condition
For at least and any nonnegative integer , the singleton set comprising the elementary abelian subgroup of order is a collection of subgroups satisfying a weak 2-subnormal replacement condition.
References
- Limits of abelian subgroups of finite p-groups by Jonathan Lazare Alperin and George Isaac Glauberman, Journal of Algebra, ISSN 00218693, Volume 203, Page 533 - 566(Year 1998): ^{Weblink for Elsevier copy}^{More info}, Theorem D