# P-solvable not implies Glauberman type for p

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., p-solvable group) neednotsatisfy the second group property (i.e., group of Glauberman type for a prime)

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## Contents

## Statement

It is possible to have a prime number and a finite group such that is -solvable (i.e., is a p-solvable group) but not a group of Glauberman type for .

In fact, this is possible for the primes .

## Related facts

- strongly p-solvable implies Glauberman type for p: In particular, this shows that for , -solvable implies Glauberman type for .
- Glauberman type not implies p-constrained
- Glauberman type not implies p-stable

## Proof

### Example at the prime two

`Further information: symmetric group:S4`

Consider the prime and the group , the symmetric group on the set . is a solvable group, and hence is -solvable. However, is not of Glauberman type for the prime two, because if we take as the -Sylow subgroup generated by , then . In this case, we have:

.

### Example at the prime three

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