P-solvable not implies Glauberman type for p
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) need not satisfy the second group property (i.e., group of Glauberman type for a prime)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about p-solvable group|Get more facts about group of Glauberman type for a prime
In fact, this is possible for the primes .
- 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
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: