# Questions tagged [automorphism-group]

For finding, constructing and proving results about automorphisms and applying them into different contexts. An automorphism is an isomorphism from an object to itself, and collectively they form a group under composition of mappings. Sometimes the only automorphism is the trivial one (the identity map), but often structures will have interesting/non-trivial automorphisms.

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### Find the order of the following elements in $Inn(S_5)$

Find the order of the following elements in Inn($S_5$). a) $\phi_{(1243)}$ So the elements in Inn($S_5$) are functions $\phi_{(1243)}: S_5 \to S_5$ via $x \to (1243)x(1243)^{-1}$. So the order of ...
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### Let $p$ be the smallest prime divisor of $|G|$ and $N\unlhd G$ s.t. $|N|=p$. Find ${\rm Aut}(N)$. [closed]

Let $p$ be the smallest prime divisor of $|G|$ and $N\unlhd G$ s.t. $|N|=p$. Find ${\rm Aut}(N)$. Attempt: We have $|N|=p$ and since $p$ is a prime number, $N$ is a cyclic subgroup and so abelian, ...
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### If $G$ is a group of order $p + 1$, then $p$ does not divide $|\text{Aut}(G)|$ ($p$ is prime and $p + 1$ isn't a prime power)

If $G$ is a group of order $p + 1$, then $p$ does not divide $|\text{Aut}(G)|$ ($p$ is prime and $p + 1$ isn't a prime power). Here is how far I got: Assume $p$ divides $|\text{Aut}(G)|$. By Cauchy's ...
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### How do subgroups of the inner automorphisms of a group look like?

I'm trying to prove the following proposition: Let $G$ be a group. Then $G$ is nilpotent iff ${\rm Inn}(G)$ is nilpotent. I've proven that if $G$ is nilpotent then ${\rm Inn}(G)$ is nilpotent as ...
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### Automorphisms of complete intersections

Suppose two hypersurfaces $H_1 = V(f_1)$ and $H_2=V(f_2)$ in $\mathbb{P}^n$ are such that $X :=H_1 \cap H_2$ is a non-singular complete intersection, i.e., the ideal of $X$ is generated by codim$(X)$ ...
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1 vote
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### A proof on automorphism groups of hypersurfaces

I am studying the paper of Matsumura & Monsky on the automorphisms of hypersurfaces and I can't understand a part of the proof of theorem 2. I restate it for clarity. For $d \geq 5$ let $H_d$ be a ...
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### Is the order of a field automorphism equal to the degree over the fixed-point subfield?

Given any field $L$ and any automorphism $f:L \to L$, one could define the fixed-point subfield $K := \{x \in L \mid f(x)=x\}$ in the obvious way. Now, suppose that $f$ has finite order $n$. Does this ...
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### Automorphism group of $\mathbb{Q}/\mathbb{Z}$ [closed]

Consider the group of all complex roots of unity, $\mathbb{Q}/\mathbb{Z}$ (where both groups are additive groups). I was wondering what its automorphism group is ?
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### Is the set of bijective functions that are invariant under a subset is a subgroup of their automorphisms?

Let $A\subseteq X$. Is $$\mathrm{Aut}(X,A) \; = \; \Big\{ \varphi:X\to X\mid \text{\varphi is bijective and \varphi(a)\in A,\ \forall\,a\in A} \Big\}$$ a subgroup of $({\rm Aut}(X),\circ)$? If not,...
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### Is the composition of two "pure" inner automorphisms of a Lie algebra a pure inner automorphism?

Given a Lie algebra $\mathfrak{g}$ and an ad-nipotent element $x$ of $\mathfrak{g}$, it can be shown that $\exp(\operatorname{ad} x)$ is a Lie algebra automorphism of $\mathfrak{g}$. Its inverse can ...
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### Automorphisms of pro-objects

Let $C$ be a small category and let $Pro(C)$ denote its pro-completion. Is it then true that $Aut(X) \cong \varprojlim_{i \in I} Aut(X_i)$ for all objects $X := \{X_i\}_{i \in I} \in Pro(C)$ ? What ...
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### When is $D_n\approx\operatorname{Aut}(D_n)$? [duplicate]

We define the group $D_n$ to be the dihedral group of order $2n$ (equivalent to the group of rotations and reflections on a regular $n$-gon) and $\operatorname{Aut}(G)$ to be the group of ...
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1 vote
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### Is $\operatorname{Hol}(D_4)$ isomorphic to a familiar group?

We define the holomorph of a group, $\operatorname{Hol}(G)$, as its semidirect product $G\rtimes _f\operatorname{Aut}(G)$. As it happens (as is shown here), $D_4\approx\operatorname{Aut}(D_4)$, and we ...
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### If $H\le G$, when can we embed $\mathrm{Aut}(H)$ into $\mathrm{Aut}(G)$?

If $H$ is a subgroup of $G$, whether an automorphism of $H$ can always be extended to an automorphism of $G$, inducing an embedding of ${\rm Aut}(H)$ into ${\rm Aut}(G)$? If not, what are some counter-...
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