# 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.

302 questions
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### How to know if an automorphism is induced by the normalizer

This is directly related to an initial question I had here. I want to followup this question with another one. Supposing I know $G\le S_n$ is a permutation group and that $C_1$ and $C_2$ are conjugacy ...
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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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### 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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### Automorphism group on a matroid

I'm trying to prove that the automorphism group on a matroid is (set-theoritically) equal to the automorphism group on its dual matroid. that is, $\ Aut(M) = Aut(M^*)$ where the automorphism group ...
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### A counterexample about inner automorphisms

Let \begin{align} \phi_x:G & \longrightarrow G\\ g & \longmapsto \phi_x(g)=x^{-1}gx \end{align} be the $x$-inner automorphism. If $G$ is a group and $H\le G$ is a characteristic subgroup ...
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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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### Automorphisms and basic algebras

Suppose I have a commutative ring $R$ and a finite-dimensional $R$-algebra $A$, and a so-called basic idempotent $e$ (an idempotent $e$ such that $eAe$ is a basic algebra of $A$). It is known that $A$ ...
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### $\sigma(a)$ is a root of the minimal polynomial for $\alpha$ over $F$? [closed]

If $K/F$ is a field extension and $\alpha \in K$ is algebraic over $F$, then for any automorphism $\sigma \in \operatorname{Aut}(K/F)$, $\sigma(a)$ is a root of the minimal polynomial for $\alpha$ ...
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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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### The automorphism group of $S_6$ is isomorphic to a semidirect prodct

On this document an outer automorphism of $S_6$ is constructed. I would like to use this construction to prove that $\mathrm{Aut}(S_6)\cong S_6\rtimes_\varphi\mathbb{Z}_2$. The idea would be to find ...
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### Automorphisms of the field of rational functions $\Bbb C(t)$

Given the field of rational functions $\Bbb C(t)$, how do we show that a certain function defines an automorphism. I ask because I read here https://nptel.ac.in/courses/111101001/downloads/problemset8....
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### A proof about Automorphism in congruence class

Suppose $gcd(m,n)=1$, and let $F :Z_n→Z_n$ be defined by $F([a])=m[a]$. Prove that $F$ is an automorphism of the additive group $Z_n$. I find it is diffcult to prove $F$ is injective and surjective. ...
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### how to check if there is an automorphism mapping between two conjugacy class

Let $G\le S_n$ be a permutation group and suppose that $C_1,C_2$ are two distinct conjugacy classes that have the same cardinality and is represented by a permutation of the same cycle-type. My ...
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### The automorphism group of the countable atomless Boolean algebra does not have ample generics

I was told that the automorphism group of the countable atomless Boolean algebra does not have ample generics. I assume that one would show this by using the Fraisse-theoretic characterizations of ...
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### Smooth covering maps and the fundamental group

Let $M$ be a smooth, connected and locally path-connected manifold, and let $\pi: \tilde{M}\to M$ be its universal cover. Let $\text{Aut}_\pi(\tilde{M})$ be the group of smooth covering ...
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### Finding the automorphism group of a graph

Let me denote the graph in the picture by $\Gamma$ and I will refer to the points as numbers $1-9$. I need to find Aut($\Gamma$). Looking at this graph, it seems that there can be a permutation on ...
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### Characteristic subgroups of Aut(G)

For a group $G$, let Aut$(G)$ denote the group of all the automorphisms of $G$. Consider the following subgroups of Aut$(G)$: Inn$(G)=$the group of inner automorphisms of $G$ IA$(G)=$ the group of ...
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### Proving Characteristic Subgroups are Transitive

I know this is exactly the same as this question. But the proof detailed uses restrictions, and I'm not familiar with that. The way I want to prove this is by the standard method of showing two sets ...
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### Problem related to semidirect product

I have a small question regarding the semidirect product. Consider a group $G$ which is the semidirect product $\mathbb{Z}_3 \ltimes (\mathbb{Z}_5 \times \mathbb{Z}_5)$ (internal semidirect product). ...
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### Conjugacy Classes in $PSL_2(\mathbb{F}_p)$

Let $p$ be an odd prime such that $2$ is not a square mod $p$. I want to determine all conjugacy classes of elements of order $p$ in $PSL_2(F_p)$. Since each such element has an eigenvalue $1$ and ...
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### Showing $x^m$ is an automorphism

Let $m,n \in \mathbb N$ and they are relatively prime and $G$ is a group with $|G|=n$. Show that $T:G \to G$ is an automorphism of $G$ where $T(x)=x^m$ First I showed it's a homomorphism. I have ...
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### Pictorial understanding for automorphism?

I know that automorphism works by mapping an element over a some ring to another different or same element over the same ring. How can we graphically understand automorphism?
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### Index of centralizer if $H\lhd G$ and $|H| < \infty$

I have read the first answer for this question Centralizer of a finite normal subgroup has finite index and I did not understood, why if $|Aut(H)| < \infty$, then $kernel = C_G(H)$ must have finite ...
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### Inner Automorphism using Transpositions

Test question that I got destroyed by. Thought about it all weekend and came up with an explanation. Today, scores back, and prof gave a very different explanation from the one I had in mind. But I'm ...
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### Group automorphism of multiplicative group of real number field

Let $\mathbb{R}$ be the real number field and $\mathbb{R}^{\times}$ be the multiplicative group of it. $\mathrm{Aut}(\mathbb{R}^{\times})$ denotes the group automorphism of $\mathbb{R}^{\times}$. [...
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### Show that $f(z)=\frac{1}{2\pi}\int\limits_0^{2\pi}f\left(\frac{e^{i\theta}+z}{1+\overline{z}e^{i\theta}}\right)d\theta$

Let $f$ be analytic on domain $\Omega$ which contains the closed unit disk $\overline{\mathbb{D}}$. Show that (a) $$f(0)=\frac{1}{2\pi}\int_0^{2\pi}f(e^{i\theta})d\theta$$ (b) Use part (a) to ...
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### A question about groups, that are isomorphic to the automorphism groups of their cycle graphs

Suppose $G$ is a finite group. Let’s call an element $g \in G$ primitive, iff $\forall h \in G$, such that $\exists n \in \mathbb{N}, h^n = g$ it is true, that $\langle h \rangle = \langle g \rangle$. ...
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### Automorphisms in the integer additive group

Consider $f_a:(\mathbb{Z},+) \rightarrow(\mathbb{Z},+), f_a(k)=ka, \forall k \in\mathbb{Z}$ the endomorphisms in the integer additive group. I have to prove that there are only two ...
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### Why are there 6 automorphisms for this graph?

o | o---o---o---o \ / o o \ / \ o o \ / \ / o / o o (I apologise for the poor diagram unable to insert an ...
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### Group isomorphism $h:(\mathbb R,+)\to (\mathbb R^+,\times)$ that is not an exponential function.

Let $\mathbb{R}^+$ denote the set of positive real numbers. Group isomorphism $h_b:(\mathbb R,+)\to (\mathbb R^+,\times)$ can be given by the exponential function: $h_b(r)=b^r$, where $b$ is a ...
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### Automorphism group of a finite group [closed]

For which finite groups $G$ is $\operatorname{Aut}(G)$ isomorphic to a (non necessarily strict) subgroup of $G$?
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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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### Meaning of stable $CP^2$

I came across the following phrase in arXiv:1903.08904 ....in order to have a stable $CP^2$ , i.e., one in which all the automorphism group is fixed... Can anyone explain to me what one means by ...
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### Automorphism of $\mathbb{C}$ over a subfield $K$ of $\mathbb{C}$ [duplicate]

Assume a finite field extension $\mathbb{C}/K$ such that $[\mathbb{C}:K]>2$. Let $\varphi \in \text{Aut}(\mathbb{C}/K)$, so $\varphi \in \text{Aut}(\mathbb{C})$ and $\varphi\vert_K=\text{id}\vert_K$...
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### Prove if the the groups $A_4$ and $S_3 \times \Bbb Z_2$ are or not are isomorphic

I'm trying to check if the groups $A_4$ and $S_3 \times \Bbb Z_2$ are or not isomorphic. How can I check if they are? I'm trying to understand how can I generally prove an isomorphism with this kind ...
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### Automorphism of a matrices ring

Let $R$ be the ring of $3 \times 3$ matrices with coefficients in $\Bbb Z_5$. For every $g \in GL_3(\Bbb Z_5)$ prove that the function $$f\colon R \rightarrow R$$ defined as $$x \mapsto g^{-1}xg$$ ...
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### Connection between sum of Graphs and their automorphism groups

can we say something about the automorphism group of a graph $G$ that has the property: $G \cong A + B$ , if we know the automorphism groups of $A$ and $B$ respectively. The $+$ is the union $\cup$ ...
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### Almost all trees have non-trivial automorphism group

In their paper Asymmetric Graphs Erdős and Rényi proved that almost all trees have non-trivial automorphism group. More specifically they showed that almost all trees contain at least one so-called ...
### Let $G,G'$ be two digraphs, show that $\phi^{-1}$ is an isomorphism
Problem So let $\phi : G \rightarrow G'$ be an isomorphism between two directed graphs. Prove that $\phi^{-1}$ is an isomorphism. Also prove if $H \leq Aut(G')$ then $\phi^{-1}H\phi\leq Aut(G)$. My ...