P-constraint is not subgroup-closed
This article gives the statement, and possibly proof, of a group property (i.e., p-constrained group) not satisfying a group metaproperty (i.e., subgroup-closed group property).
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- p-constraint is not quotient-closed
- p-solvable implies p-constrained
- p-constrained not implies p-solvable
- Constrained for a prime divisor implies not simple non-abelian
Let be the wreath product of Z2 and A5 defined as the wreath product with base group cyclic group:Z2 and acting group alternating group:A5, where we use the natural permutation action of the acting group on a set of five elements. More explicitly, is the external semidirect product of elementary abelian group:E32 and alternating group:A5 where the latter acts on the former by coordinate permutations induced by the permutations on a set of five elements.
The group has order
We note that:
- is -constrained: Indeed, is trivial, and is the base of the semidirect product, i.e., a normal subgroup isomorphic to elementary abelian group:E32. In particular, this is contained in any -Sylow subgroup , so is also the normal subgroup that forms the base of the semidirect product. The subgroup is a self-centralizing normal subgroup, because it is an abelian normal subgroup and the induced action by the quotient is faithful. Thus, we get the condition .
- has a subgroup isomorphic to alternating group:A5, namely the non-normal part of its realization as a semidirect product. This subgroup is a simple non-abelian group with dividing its order (i.e., 2 divides 60) and hence, by Fact (1), is not a -constrained group.