# Isomorphic to inner automorphism group not implies centerless

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., group isomorphic to its inner automorphism group) need not satisfy the second group property (i.e., centerless group)
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## Statement

There can exist a group  such that the Center (?) of  is nontrivial, but  is isomorphic to the quotient group  (which is also the Inner automorphism group (?) of ).

## Proof

### Example of a generalized dihedral group

Further information: generalized dihedral group of 2-quasicyclic group

Let  be the group of all  roots of unity among the complex numbers, under multiplication, for all . Let  be the semidirect product of  with a cyclic group of order two acting by the inverse map. In other words,  is the generalized dihedral group corresponding to the abelian group . Then, we have:

• The center  of  is a group of order two inside , namely, the elements : Clearly, no element outside  is in the center, because any element outside  acts on  by the inverse map under conjugation. This leaves elements inside . Any central element of  must be fixed by conjugation by elements outside , namely, the inverse map. But there are only two elements of  that are fixed by the inverse map, namely .
• The quotient  is isomorphic to : Note that  is isomorphic to , via the map sending  to  (the map is well-defined because both elements in the coset of , namely  and , get sent to the same element: ). Further, the inverse map commutes with this map, so it extends to an isomorphism between  and .