Mann's replacement theorem for subgroups of prime exponent

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This article defines a replacement theorem
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Statement

Statement in terms of strong normal replacement

Suppose p is a prime and G is a finite p-group. In other words, G is a group of prime power order. Suppose G contains a subgroup H of order p^k, with k < p, and exponent p. In other words, H is a small group of prime exponent. Then, G contains a normal subgroup of order p^k and prime exponent, which is contained in the normal closure of H in G.

In other words, the collection of groups of order p^k and exponent p is a Collection of groups satisfying a weak normal replacement condition (?). Hence, it is also a Collection of groups satisfying a weak normal replacement condition (?).

Statement in terms of universal congruence condition

Suppose p is a prime and G is a finite p-group. In other words, G is a group of prime power order. Suppose G contains a subgroup H of order p^k, with k < p, and exponent p. In other words, H is a small group of prime exponent.

Then, the number of subgroups of G of order p^k and exponent p is congruent to 1 mod p.

In other words, the collection of groups of order p^k and exponent p is a Collection of groups satisfying a universal congruence condition (?).

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