Open main menu

Groupprops β

Hall not implies order-conjugate

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.

Related facts

Facts used

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.