Open main menu

Groupprops β

Order-conjugate and Hall not implies order-dominating

Statement

There can exist a finite group G with a Hall subgroup H such that the order and index of H in G are relatively prime, and such that H is conjugate to any subgroup of G of the same order as H, but such that there exists a subgroup math>K</math> of G such that the order of K divides the order of H, but K is not contained in H.

Proof

Example of the alternating group of degree five

Further information: alternating group:A5, subgroup structure of alternating group:A5

Suppose G is the alternating group on the set S = \{ 1,2,3,4,5 \}. Suppose H is the subgroup of G that is the alternating group on \{ 1,2,3,4 \} (it is isomorphic to alternating group:A4). In other words, H is the stabilizer of the point \{ 5 \}. Then, H is a Hall subgroup of G. Further, H is order-conjugate in G: the subgroups of the same order as H are precisely the stabilizers of points in S, and these are conjugate to H by suitable 5-cycles.

On the other hand, consider the subgroup K:

K := \{ (), (1,2,3), (1,3,2), (1,2)(4,5), (2,3)(4,5), (1,3)(4,5) \}.

K is a group of order six, isomorphic to the symmetric group of degree three. However, K is not contained in any conjugate of H, because any conjugate of H stabilizes some element, and K does not stabilize any element.