# Order-conjugate and Hall not implies order-dominated

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 $20$.