# Glauberman type not implies p-constrained

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., group of Glauberman type for a prime) neednotsatisfy the second group property (i.e., p-constrained group)

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## Statement

It is possible to have a prime number and a finite group such that is a group of Glauberman type for but is not a p-constrained group.

## Proof

### Example of the special linear group

`Further information: special linear group:SL(2,5), subgroup structure of special linear group:SL(2,5)`

Let and . Then, is cyclic of order two, and is a simple group isomorphic to alternating group:A5. If is a -Sylow subgroup of , then is isomorphic to a quaternion group and , so . Thus, is not -constrained.

On the other hand, , so , so is of Glauberman type with respect to the prime .