# 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.

301 questions
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### Order of automorphism group of cyclic group

Let $G$ be a cyclic group of order $m$. What is the order of $\text{Aut}(G)$? I want to know the proof as well (elementary if possible). I would still accept the proof if one answers with $m = p$, a ...
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### If a function has an infinite amount of automorphisms would that imply that it is periodic?

Let $\phi(x)$ be continuous and differentiable everywhere and its automorphisms be bijections $\gamma_{n}$ such that $\phi(\gamma_{n}(x))=\phi(x)$. If the order of $Aut(\phi)$ is infinite, would that ...
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### Automorphism group of the projective unitary group PU(N) and SO(N)

I would like to determine the automorphism group of the projective unitary group $G=PU(N)=PSU(N)$ and $G=SO(N)$. We also knew that $$0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0.$$...
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### Variety with nonalgebraic automorphism group

In Algebraic Geometry - A First Course by Joe Harris, an example in section 10 on algebraic groups reads that there exist varieties whose automorphism group is not an algebraic group. To give an ...
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### What can we say about the equivalence relation defined such that $G \sim H \Longleftrightarrow Aut(G) \simeq Aut(H)$?

Given two finite graphs $G, H$, I say that $G \sim H$ if and only if $Aut(G) \simeq Aut(H)$. We define $[G] = \{H \mid Aut(G) \simeq Aut(H) \}$ for any graph $G$. Due to theorems by Erdos and others ...
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### Groups which can not occur as automorphism group of a group

Consider the following natural question: Given a finite group $H$, does there exists a finite group $G$ such that $\mathrm{Aut}(G)\cong H$? In short, does any finite group occurs as the ...
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### What is an example of a power automorphism of a group that isn't a universal power automorphism?

A power automorphism maps every subgroup of a group to within itself, with equality if the group is finite. More specifically, for a subgroup $H$ of $G$, a power automorphism $f$ has $f(H) \subseteq H$...
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### What is the smallest poset with automorphism group $C_n$?

I've recently been interested in finding small finite posets (and thereby finite $T_0$ topologies) with a given automorphism group. I came upon the paper of Barmak and Minian in which they provide an ...
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### Examples of a group $G$ with a non-trivial homomorphism $f:G \to Z(G)$

I recently learned that if $f: G \to Z(G)$ is a homomorphism of $G$ to its center, then $g:G \to G$ defined as $g(x)=f(x)x$ is an endomorphism of $G$. I am having trouble thinking of examples of a (...
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### Gordon Royle's 21-vertex 21-automorphism graph

OEIS A080803 lists the minimal number of vertices $a(n)$ needed to support an undirected graph whose automorphism group has order $n$. The MathWorld page on graph automorphisms links to this sequence ...
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### Is there a way to describe the structure of $Aut(UT(3, p))$?

Is there a way to describe the structure of the automorphism group of $$C_{p}^2 \rtimes C_p \cong \langle x, y, z | [x,y]=z, [x,z]=[y,z]=x^p=y^p=z^p=e \rangle \cong UT(3, p)?$$ Here $p$ is an odd ...
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### Is there a way to classify all metabelian finite groups $G$, such that $\operatorname{Aut}(G) \cong G$?

Is there a way to classify all metabelian finite groups $G$, such that $\operatorname{Aut}(G) \cong G$? I know that the trivial group is the only abelian group that satisfies this condition. I also ...
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### Subgroup of coprime order with automorphism group is contained in center of group

I'm studying for a qualifying exam in algebra and I've come across the following problem: Let $G$ be a finite group with a subgroup $N$. Let $Aut(G)$ be the group of automorphisms of $G$. Prove ...
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### The Automorphism Group of Free Quandles?

The following definitions are quoted from this article: My question is, do we know anything description the automorphism group of free quandles with relative smaller number of generating set $S$? For ...
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### Determine invariant subfield and order in $\operatorname{Aut}(L)$

I'm trying to determine some explicit invariant subfields of $L=\mathbb{Q}(X)$. We have defined $\sigma_i\in$ Aut$(L)$ with $$\sigma_1(X)=1/X,\ \sigma_2(X)=1-X.$$ We are now asked to determine the ...
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### Automorphism group of ($\mathbb{R}^{\times}, \cdot$)

Let ($\mathbb{R}^{\times}, \cdot$) be the multiplicative group of non-zero real numbers. I want to find the automorphism group of ($\mathbb{R}^{\times}, \cdot$). My guess is that it is the group of ...
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### Holomorph of a group $G$, then the automorphism of $G$ are inner automorphisms
In my course notes of algebra it says: Let $G$ be a group. Then $\mathrm{Aut}(G)$ acts on $G$ in a natural way through automorphisms. This allows us to consider $A:= G \rtimes \mathrm{Aut}(G)$. In ...
### Boundary $\partial F_n$ of a free group $F_n$
I am reading "The Tits alternative for $\operatorname{Out}(F_n)$ I: Dynamics of exponentially-growing automorphisms" by Mladen Bestvina et al. for my Master's thesis. On page 526, the following part ...