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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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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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Subgroup of ${\rm Aut}\,(\widehat{\mathbb{Z}})$

Ribes and Zalesskii Corollary 4.4.8 show that the group of continuous automorphisms of $\widehat{\mathbb{Z}}$ satisfies ${\rm Aut}\,(\widehat{\mathbb{Z}})\cong\mathbb{Z}_2\times\frac{\mathbb{Z}}{2\...
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Galois group of a quintic with 3 real roots. How to conclude that there's one cycle of order 5?

I understand perfectly the argument making use of Cauchy's theorem, which I'll lay down for clarity's sake: take $p(x)$ of degree 5 irreducible over $\mathbb{Q}$. Let $K$ be the root field of $p(x)$ ...
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1answer
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The order of conjugate subgroups.

Let $G$ be a group, and let $H$ be a subgroup of $G$. Then, for any fixed $g\in G$, i know that $H\cong gHg^{-1}$ by inner-automorphism. I have some question about the 'order': (1) Is it true that $...
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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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extending automorphisms in a tower of fields

Say we have the following tower of fields: $\mathbb{Q} \subset \mathbb{Q}(\sqrt{5}) \subset \mathbb{Q}(\sqrt[4]{5})$. For ease, we can write $\mathbb{Q} = F, \mathbb{Q}(\sqrt{5}) = E, \mathbb{Q}(\sqrt[...
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Preserving relations and operations in an automorphism

An automorphism is an isomorphism of an algebraic structure with itself. Let's review an isomorphism $i$ of two linearly ordered groups $G(+,<)$ and $F(*,<)$. $i$ is linear order-preserving ...
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1answer
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Structure of the outer automorphism group of $D_n(q)$

From the ATLAS, I know that the outer automorphism group of the Chevalley group $D_n(q)$, $q=p^f$ for some prime $p$ and some $n$ even and $n>4$, is a semidirect product of three groups, $(C_d \...
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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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1answer
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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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$G$ non-centerless group. How is $Z(\operatorname{Aut}(G))$ made?

Let $G$ be a (possibly infinite) non-centerless group, i.e. such that $Z(G) \ne \lbrace e \rbrace$. Left and right multiplications establish the subgroups $\Theta:=\lbrace \theta_a \mid a \in G \...
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Automorphisms of a particular abelian group

I am trying to solve Exercise 7.9 of An Introduction to K-theory of C*-algebras, It remains for me to show that the Automorphisms of $\mathbb{Q}\oplus\mathbb{Q}$ which send $ \{(x,y) \in \mathbb{Q}\...
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Internal Semidirect with Factors Isomorphic to “Outside” Groups

Here is a conjecture of mine: If $G$ is the internal semidirect product of $N \unlhd G$ and $Q \le G$, and $\phi_1 : N' \to N$ and $\phi_2 : Q' \to Q$ are isomorphisms, then there is some $\theta :...
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Sequential continuity of automorphisms of number fields

If I have some Galois number field $K$, a sequence of elements $(x_n)_{n=1}^{\infty}\subset K$ converging to some $z\in K$, and an automorphism $\sigma$ of K, when does $\lim\limits_{n\to\infty}(\...
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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 ...
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Automorphisms, order and degree uniquely determine a regular graph

Does Automorphism group combined with the order and degree uniquely determine any regular graph? What about any (non-regular) graph? I think yes, because the automorphism contain within them the ...
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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 ...