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

From Groupprops

## Contents

## Statement

### Hands-on statement

Suppose is a prime number and is a finite -group. Suppose is an abelian subgroup of of order , where and exponent dividing . Then, there exists an abelian normal subgroup of contained in the normal closure of such that has order and exponent dividing .

### Statement in terms of normal replacement condition

Suppose is a prime number and . Then, the collection of abelian groups of order and exponent dividing is a Collection of groups satisfying a strong normal replacement condition (?).

## Related facts

- Congruence condition on number of abelian subgroups of small prime power order and bounded exponent for odd prime
- Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime
- Jonah-Konvisser congruence condition on number of elementary abelian subgroups of small prime power order for odd prime
- Elementary abelian-to-normal replacement theorem for large primes (this is a weaker version that is superseded by the result on this page and the results of Jonah and Konvisser)
- Mann's replacement theorem for subgroups of prime exponent

## References

### Journal references

- Paper:Glaubermanexistenceofnormalinfinitepgroup
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