# Hall not implies order-conjugate

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., Hall subgroup) need not satisfy the second subgroup property (i.e., order-conjugate subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about Hall subgroup|Get more facts about order-conjugate subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property Hall subgroup but not order-conjugate subgroup|View examples of subgroups satisfying property Hall subgroup and order-conjugate subgroup

## Statement

There can exist a finite group $G$ and Hall subgroups $H, K$ of $G$ of the same order that are not conjugate in $G$.

## Proof

The proof follows from either of facts (1), (2) or (3). Fact (1) gives examples where the two Hall subgroups are automorphic subgroups but are not conjugate (in other words, there is an outer automorphism sending one Hall subgroup to the other, but no inner automorphism doing it). Fact (2) gives examples where the two Hall subgroups are isomorphic but there is no automorphism sending the first to the second. Fact (3) gives examples where the two subgroups have the same order but are not isomorphic, and hence cannot be conjugate.