Abelian-to-normal replacement theorem for prime exponent

This article defines a replacement theorem
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Statement

Suppose $P$ is a finite Group of prime exponent (?): group of prime power order, say $p^{r}$, and with exponent $p$ (so every element has order $p$). Suppose $A$ is an abelian subgroup of order $p^{n}$, and nilpotency class at most $p+1$.

Then, there exists an Abelian normal subgroup (?) $B$ of $P$ such that:

• $B$ is contained in the normal closure of $A$ in $P$
• $B$ has the same order (i.e., $p^{n}$) as $A$

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 copyMore info}, Theorem B