Center 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., center) does not always 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
- Center not is fully invariant in class two p-group
- Center not is normality-preserving endomorphism-invariant: We can use the same idea, but need to impose some more constraints on the generic example.
Clearly, is the center of .
Now consider the endomorphism of which composes the projection onto with the isomorphism from to . This endomorphism does not send to within itself, and hence, is not a fully characteristic subgroup of .
Thus, the center is not fully invariant.
The particular example can be obtained from the generic example above by setting to be a cyclic group of order 2 and (the symmetric group on three letters). in this case is the 2-element subgroup generated by a transposition.
|Group part||Subgroup part||Quotient part|
|Center of central product of D8 and Z4||Central product of D8 and Z4||Cyclic group:Z4||Klein four-group|
|Center of direct product of D8 and Z2||Direct product of D8 and Z2||Klein four-group||Klein four-group|