Glauberman's abelian-to-normal replacement theorem for bounded exponent and half of prime plus one

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Statement

Hands-on statement

Suppose p is a prime number and P is a finite p-group. Suppose A is an abelian subgroup of P of order p^k, where k \le (p + 1)/2 and exponent dividing p^d. Then, there exists an abelian normal subgroup B of P contained in the normal closure of A such that B has order p^k and exponent dividing p^d.

Statement in terms of normal replacement condition

Suppose p is a prime number and 0 \le d \le k \le (p + 1)/2. Then, the collection of abelian groups of order p^k and exponent dividing p^d is a Collection of groups satisfying a strong normal replacement condition (?).

Related facts

References

Journal references