groups with abelian inner automorphism group
and . Representatives of the cosets of are ; these define a group of order that is isomorphic to the Klein -group (http://planetmath.org/Klein4Group) . Thus the central quotient is abelian, but the group itself is not.
However, if the central quotient is cyclic, it does follow that is abelian. For, choose a representative in of a generator for . Each element of is thus of the form for . So given ,
where the various manipulations are justified by the fact that the and that powers of commute with themselves.
Finally, note that if is non-trivial, then is nonabelian, for nontrivial implies that for some , conjugation by is not the identity, so there is some element of with which does not commute. So by the above argument, , if non-trivial, cannot be cyclic (else would be abelian).
|Title||groups with abelian inner automorphism group|
|Date of creation||2013-03-22 17:25:30|
|Last modified on||2013-03-22 17:25:30|
|Last modified by||rm50 (10146)|