groups with abelian inner automorphism group

The inner automorphism group Inn(G) is isomorphicPlanetmathPlanetmathPlanetmath to the central quotient of G, G/Z(G). if Inn(G) is abelianMathworldPlanetmath, one cannot conclude that G itself is abelian. For example, let G=𝒟8, the dihedral groupMathworldPlanetmath of symmetriesPlanetmathPlanetmath of the square.


and Z(G)={1,r2}. Representatives of the cosets of Z(G) are {1,r,s,rs}; these define a group of order 4 that is isomorphic to the Klein 4-group ( V4. Thus the central quotient is abelian, but the group itself is not.

However, if the central quotient is cyclic, it does follow that G is abelian. For, choose a representative x in G of a generatorPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath for G/Z(G). Each element of G is thus of the form xaz for zZ(G). So given g,hG,


where the various manipulations are justified by the fact that the ziZ(G) and that powers of x commute with themselves.

Finally, note that if Inn(G) is non-trivial, then G is nonabelianPlanetmathPlanetmathPlanetmath, for Inn(G) nontrivial implies that for some gG, conjugationMathworldPlanetmath by g is not the identityPlanetmathPlanetmathPlanetmathPlanetmath, so there is some element of G with which g does not commute. So by the above argument, Inn(G), if non-trivial, cannot be cyclic (else G would be abelian).

Title groups with abelian inner automorphism group
Canonical name GroupsWithAbelianInnerAutomorphismGroup
Date of creation 2013-03-22 17:25:30
Last modified on 2013-03-22 17:25:30
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 7
Author rm50 (10146)
Entry type Topic
Classification msc 20A05