# Epicenter not is fully invariant

This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., epicenter) doesnotalways satisfy a particular subgroup property (i.e., fully invariant subgroup)

View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions

## Statement

It is possible to have a group such that the epicenter of is not a fully invariant subgroup of .

## Proof

`Further information: nontrivial semidirect product of Z4 and Z4, subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4, subgroup structure of nontrivial semidirect product of Z4 and Z4`

Suppose is the nontrivial semidirect product of Z4 and Z4, given explicitly as follows, where denotes the identity element:

As we can see from the subgroup structure of , the biggest quotient of that is a capable group is dihedral group:D8, and this arises as the quotient of by the subgroup , which is a subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4. Thus, is the epicenter of .

is not fully invariant in . For instance, it is not invariant under the endomorphism with kernel that sends to .