Hall not implies order-conjugate

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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)
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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

Similar facts about Hall subgroups

Opposite facts about Hall subgroups

Facts used

  1. Hall not implies automorph-conjugate
  2. Hall not implies isomorph-automorphic
  3. Hall not implies order-isomorphic

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.