Order-conjugate 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 subgroup) need not satisfy the second subgroup property (i.e., order-dominated subgroup)
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EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property order-conjugate subgroup but not order-dominated subgroup|View examples of subgroups satisfying property order-conjugate subgroup and order-dominated subgroup
It is possible to have a finite subgroup of a group such that is conjugate to any subgroup of of the same order as , but such that there exists a subgroup of whose order is a multiple of the order of , but such that is not conjugate to any subgroup of .
Example of the alternating group of degree five
Suppose is the alternating group on the set . Suppose is the subgroup of given by:
is conjugate to any other subgroup of of the same order.
On the other hand, suppose is the stabilizer of in . Then, is the alternating group on and is isomorphic to the alternating group of degree four. The order of is , which is a multiple of the order of . However, no conjugate of is contained in , since does not stabilize any element, but every conjugate of stabilizes some element.