Center not is endomorphism kernel
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., endomorphism kernel)
View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions
Further information: subgroup structure of quaternion group
The simplest example is where is the quaternion group, so is the center of quaternion group and the quotient group is isomorphic to the Klein four-group. has no subgroup isomorphic to the Klein four-group.