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].
Questions tagged [complete-groups]
7 questions
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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 groups have even order (a well known fact)
2)…
Chain Markov
- 16,012
5
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1 answer
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 about this problem, if $G$ is a group then let…
Bhaskar Vashishth
- 11,677
5
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1 answer
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 same time?
Motivation behind this question:
Both…
Chain Markov
- 16,012
3
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1 answer
Does $G$ centerless and $G\cong \operatorname{Aut}(G)$ imply $G$ complete
I will take as definition outer automorphisms as any automorphism that is not inner. We say that a group $G$ is complete if it is centerless and $G$ admits no outer automorphisms.
As a fact, $G$ being complete implies that $G\cong Aut(G)$. I’ve been…
Meow
- 65
3
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1 answer
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 groups, we have $G = f(N)L$ for some $L \unlhd G$…
Pratina
- 149
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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 homomorphism $\hat{\phi}: \mathbb{Z}_p \rightarrow M $ is…
user99383532
- 73
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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 $n = 1$ it is not true, however if $n = 2$, then $…
Chain Markov
- 16,012