Order-conjugate and Hall 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 Hall subgroup) need not satisfy the second subgroup property (i.e., order-dominated subgroup)
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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 .