# Questions tagged [outer-automorphisms]

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### Can $\text{Aut}(G)$ be extended to contain $G$?

Let $G$ be a group (finite, say) with center $Z$. The automorphism group $\text{Aut}(G)$ sits in a short exact sequence $$1 \to G/Z \to \text{Aut}(G) \to \text{Out}(G) \to 1.$$ So when $Z\neq 1$, as ...
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### Explicit formula for general group extension in terms of cartesian product set

According to Wikipedia and ncat lab general group extensions $$N\rightarrow G\rightarrow Q$$ are classified by a group homomorphism $$\rho: Q\rightarrow \operatorname{Out}(N)$$ subject to a constraint ...
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### Morse theory on outer space via the lengths of finitely many conjugacy classes

Let $F_n$ be the free group on letters $\{x_1,\ldots,x_n\}$ and let $X_n$ be the (reduced) outer space of rank $n$. Points of $X_n$ thus correspond to pairs $(G,\mu)$, where $G$ is a finite connected ...
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### Outer automorphisms of finitely generated linear groups

Is there an example of a finitely generated subgroup $\Gamma \subset \mathrm{GL}_n(\mathbb{C})$ such that the group of outer automorphisms $\mathrm{Out}(\Gamma)$ contains finite subgroups of unbounded ...
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### Finitely generated nilpotent groups with hyperbolic automorphisms

$\DeclareMathOperator\Out{Out}\DeclareMathOperator\GL{GL}$ Let $G$ be a finitely generated nilpotent group. We call an automorphism of $G$ hyperbolic if the induced automorphism of the free part of ...
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### Does $O(4,\mathbb{Q})$ have an exceptional outer automorphism?

Does the orthogonal group $O(4,\mathbb{Q})$ have an exceptional outer automorphism analogous to that of its subgroup, the Coxeter/Weyl group $W(F_4)$?
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### Does Aut(G) → Out(G) always split for a compact, connected Lie group G?

The outer automorphism group of a topological group $G$ is constructed by the short exact sequence  1\longrightarrow \operatorname{Inn}(G) \longrightarrow \operatorname{Aut}(G) \longrightarrow \...
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### Groups with trivial outer automorphism group and prescribed center?

Given an arbitrary abelian group $A$, can we find a group $G$ such that $\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G)=1$, and $Z(G)\simeq A$? Why is this interesting? Given a group $G$, we have ...
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### Extending an automorphism to an inner one

Let $D$ be a division ring. I have in mind the following result. Theorem. For every automorphism $f$ of $D$, there is a division ring $E$ extending $D$ such that $f$ extends to an inner automorphism ...
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### Automorphisms of the Lie algebras $\mathfrak{sl}(2,R)$ and $\mathfrak{su}(2)$

I would like to know about the literature concerning the group of outer automorphisms of the Lie algebra $\mathfrak{sl}(2,R)$. This question is addressed in different places in a contradictory way. In ...
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### What happens when you internalize outer automorphisms?

Given a finitely presented group $G = (Gen|Rel)$, we have a set of inner automorphisms $\{ \phi_a(x) = axa^{-1} | a \in G\}$. Defining the set of outer automorphisms to be those automorphisms of $G$ ...
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