Order-conjugate and Hall 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 Hall subgroup) need not satisfy the second subgroup property (i.e., order-dominating Hall subgroup)
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There can exist a finite group with a Hall subgroup such that the order and index of in are relatively prime, and such that is conjugate to any subgroup of of the same order as , but such that there exists a subgroup math>K</math> of such that the order of divides the order of , but is not contained in .
Example of the alternating group of degree five
Suppose is the alternating group on the set . Suppose is the subgroup of that is the alternating group on (it is isomorphic to alternating group:A4). In other words, is the stabilizer of the point . Then, is a Hall subgroup of . Further, 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.