2 edited title

1

# Are there any 2 groups with Complete Holomorphs Other than $C^2_2$ and $C^4_2$?

The title says it. I'm reviewing some group theory concepts I haven't touched in quite a while, since I don't teach abstract algebra in my current position, and could not find the answer to this question. I've searched on google and found some papers that discuss other types of groups, but not 2-groups. I know that the holomorph of all non-trivial finite abelian groups of odd order are complete groups, due to a theorem by Miller(1908), where complete means trivial center and all automorphisms are inner. Also, since the holomorph of a finite abelian group is the direct product of the holomorphs of its Sylow subgroups, again Miller(1903?), does that imply the following:

Let G be an even ordered abelian group. If the holomoprh of the Sylow 2-subgroup of G is complete then the holomorph of G is complete.

As an added note, the automorphism group of $C^n_2$ is isomorphic to $PSL(n,2)$ since $C^n_2$ can be thought of as an $n$ dimensional vector space over the finite field $Z_{2}$. In the case where $n$ = 4, $PSL(4,2)$ is isomorphic to $A_{8}$. Is this connected to the holomorph being complete? Is the answer known for $C^n_2$, if not for all 2-groups?

Thanks