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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Complete groups
Recently, I have been studying complete groups, and I know that if a group $N$ is complete then for every short exact sequence $1 \rightarrow N \xrightarrow{f} G \xrightarrow{g} H \rightarrow 1$ of ...
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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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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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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 $...
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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 ...
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If $G$ is complete, then the holomorph of $G$ is isomorphic to $G\times G$.
Question- If $G$ is complete, then the holomorph of $G$ is isomorphic to $G\times G$.
I am studying semidirect products for the first time, and in some notes I found this exercise. As far as I know ...
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