Order-conjugate 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 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 subgroup H of a group G such that H is conjugate to any subgroup of G of the same order as H, but such that there exists a subgroup K of G whose order is a multiple of the order of H, but such that H is not conjugate to any subgroup of K.

Related facts

Proof

Example of the alternating group of degree five

Further information: alternating group:A5, subgroup structure of alternating group:A5

Suppose G is the alternating group on the set \{ 1,2,3,4,5 \}. Suppose H is the subgroup of G given by:

H := \{ (), (1,2,3), (1,3,2), (1,2)(4,5), (2,3)(4,5), (1,3)(4,5) \}.

H is conjugate to any other subgroup of G of the same order.

On the other hand, suppose K is the stabilizer of \{ 5 \} in G. Then, K is the alternating group on \{ 1,2,3,4 \} and is isomorphic to the alternating group of degree four. The order of K is 12, which is a multiple of the order of H. However, no conjugate of H is contained in K, since H does not stabilize any element, but every conjugate of K stabilizes some element.