# Order-conjugate and Hall not implies order-dominated

From Groupprops

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) neednotsatisfy the second subgroup property (i.e., order-dominated subgroup)

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## Statement

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

## Proof

Consider the projective special linear group . Let be a -Hall subgroup of (it turns out that such a is a dihedral group of order twenty) and be a subgroup of isomorphic to the alternating group of degree five. Then, we have the following:

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