Order-conjugate and Hall not implies order-dominated
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Revision as of 04:30, 28 February 2009 by Vipul (talk | contribs) (New page: {{subgroup property non-implication| stronger = order-conjugate Hall subgroup| weaker = order-dominated subgroup}} ==Statement== It is possible to have a finite group <math>G</math>,...)
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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EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property order-conjugate Hall subgroup but not order-dominated subgroup|View examples of subgroups satisfying property order-conjugate Hall subgroup and order-dominated subgroup
Statement
It is possible to have a finite group , an order-conjugate Hall subgroup
of
, and a subgroup
of
whose order is a multiple of the order of
such that no conjugate of
is contained in
.
Proof
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
.