Order-conjugate not implies order-dominating
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-dominating subgroup)
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We can have a group with an order-conjugate subgroup (i.e., a subgroup that is conjugate to any other subgroup of the same order) that is not an order-dominating subgroup: in other words, there exists a subgroup of whose order divides the order of , but such that is not contained in any conjugate of .
Example of the alternating group of degree five
Further information: alternating group:A5
Suppose is the alternating group on the set . Suppose is the subgroup of that is the alternating group on . In other words, is the stabilizer of the point . Then, is order-conjugate in : the subgroups of the same order as are precisely the stabilizers of points in , and these are conjugate to by suitable -cycles.
On the other hand, consider the subgroup :
is a group of order six, isomorphic to the symmetric group of degree three. However, is not contained in any conjugate of , because any conjugate of stabilizes some element, and does not stabilize any element.