Isomorphic to inner automorphism group not implies centerless

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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 .