Abelian-to-normal replacement theorem for prime exponent
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This article defines a replacement theorem
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Statement
Suppose is a finite Group of prime exponent (?): group of prime power order, say , and with exponent (so every element has order ). Suppose is an abelian subgroup of order , and nilpotency class at most .
Then, there exists an Abelian normal subgroup (?) of such that:
- is contained in the normal closure of in
- has the same order (i.e., ) as
Related facts
- Mann's replacement theorem for subgroups of prime exponent
- p-group is either absolutely regular or maximal class or has normal subgroup of exponent p and order p^p
- Group of exponent p and order greater than p^p is not embeddable in a maximal class group
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 B