Order-conjugate and Hall not implies order-dominated

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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., order-conjugate Hall subgroup) need not satisfy the second subgroup property (i.e., order-dominated subgroup)
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Statement

It is possible to have a finite group G, an order-conjugate Hall subgroup H of G, and a subgroup K of G whose order is a multiple of the order of H such that no conjugate of H is contained in K.

Proof

Consider the projective special linear group G = PSL(2,61). Let H be a \{2,5 \}-Hall subgroup of G (it turns out that such a H is a dihedral group of order twenty) and K be a subgroup of G isomorphic to the alternating group of degree five. Then, we have the following:

  • H is conjugate to any other subgroup of the same order: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
  • The order of H divides the order of K: Indeed, H has order 20 and K has order 60.
  • No conjugate of H is contained in K: In fact, the alternating group of degree five contains no subgroups of order 20.