# P-constraint is not quotient-closed

From Groupprops

This article gives the statement, and possibly proof, of a group property (i.e., p-constrained group)notsatisfying a group metaproperty (i.e., quotient-closed group property).

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## Contents

## Statement

We can have a finite group , a prime number , and a normal subgroup of such that is a p-constrained group but is *not* a p-constrained group.

## Related facts

- p-constraint is not subgroup-closed
- p-solvable implies p-constrained
- p-constrained not implies p-solvable
- Constrained for a prime divisor implies not simple non-abelian

## Facts used

## Proof

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

Let .

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 .
- is not -constrained: Since is the normal subgroup that forms the base of the semidirect product, the quotient group is alternating group:A5, which is a simple non-abelian group, with dividing its order, hence by Fact (1) is not -constrained.