# Mann's replacement theorem for subgroups of prime exponent

This article defines a replacement theorem

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## Contents

## Statement

### Statement in terms of strong normal replacement

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

In other words, the collection of groups of order and exponent 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 is a prime and is a finite -group. In other words, is a group of prime power order. Suppose contains a subgroup of order , with , and exponent . In other words, is a *small* group of prime exponent.

Then, the number of subgroups of of order and exponent is congruent to 1 mod .

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