Questions tagged [complete-groups]

A complete group is a centerless group that has only inner automorphisms ($\textrm{Aut}(G) = \textrm{Inn}(G)$). Equivalently a centerless group is complete iff $\textrm{Aut}(G)$ is isomorphic to $G$. To be used with the tag [group-theory].

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185 views

Does there exist a non-trivial group that is both perfect and complete?

A group $G$ is called perfect iff $G’ = G$. A group $G$ is called complete iff $Z(G) = \{e\}$ and $Aut(G) \cong G$. Does there exist a non-trivial group $G$, that is both perfect and complete at the ...
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1answer
25 views

Injection of $\mathbb{Z}$ into a p-adically complete abelian group

If $M$ is a p-adically complete abelian group, so that it's a $\mathbb{Z}_p$-module, and we have an injective homomorphism $\phi: \mathbb{Z} \hookrightarrow M$, is it true that the induced ...
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1answer
63 views

For what $n \in \mathbb{N}$ is $ \operatorname{Hol}(C_2^n)$ complete?

For what $n \in \mathbb{N}$ is $ \operatorname{Hol}(C_2^n)$ complete? Here $ \operatorname{Hol}$ stands for holomorph, and $C_2^n$ stands for direct product of $n$ isomorphic copies of $C^2$. It for $...
7
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1answer
108 views

Does every finite non-trivial complete group have even order?

Does every finite non-trivial complete group have even order? I checked three well known classes of complete groups, and this statement is true for them all: 1) Symmetric groups: All symmetric ...
4
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1answer
150 views

A question about holomorph of groups.

Question- If $G$ is complete, then the holomorph of $G$ is isomorphic to $G$×$G$. I am studying semidirect products for the first time, and in some notes i found this exercise. As far as i know about ...