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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A field automorphism that is the identity outside a subfield

I was reading Lemma 2 of Daniel Lascar' The group of automorphisms of the field of complex numbers leaving fixed the algebraic numbers is simple whose statement is the following: Assume $g \in \text{...
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Prove if $G$ is a finite nonabelian $p$-group, then $p^2\mid |{\rm Aut}(G)|$. [duplicate]

Prove if $G$ is a finite nonabelian $p$-group, then $p^2\mid|{\rm Aut}(G)|$. Suppose $|G|=p^m, m\in \mathbb{N}$. A fact I know about $p$-groups: Since $G$ is a $p$-group, $\forall i\leq m \space \...
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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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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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Inner automorphisms of Pauli strings in the unitary subgroup of matrices

Statement What is the set $\mathcal{T}_n$ of matrices $T \in GL(2^n)$ such that for all Pauli strings $P \in \mathcal{P}_n=\{\otimes_{i=1}^n \sigma_{m_i}\mid m\in \{1,2,3\}^n, \sigma_{m_i} \text{ ...
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Twisted graded algebras [J.J. Zhang]

I am studying the article by James J. Zhang Twisted Graded Algebras And Equivalences of Graded Categories, I have consulted to know more about the topics they address such as twisted algebras, torsion ...
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Structural stability of Arnold's Cat Map

In Wikipedia it says that "hyperbolic automorphisms of the torus, such as the Arnold's cat map, are structurally stable", so I am trying to understand why this is true. What $C^1$-small ...
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Let $G_1,G_2$ be groups. $G_1 \cong G_2 \implies\operatorname{Aut}(G_1) \cong \operatorname{Aut}(G_2)$.

Let $G_1,G_2$ be groups. $$ G_1 \cong G_2 \implies \operatorname{Aut}(G_1) \cong \operatorname{Aut}(G_2). $$ My solution : Since $G_1 \cong G_2$ there is an isomorphism $f : G_1 \to G_2$. Denote $\phi:...
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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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Automorphism group of the projective line $\mathbb{P}^1(\mathbb{Z})$ [duplicate]

Consider ``the'' projective line over the integers (which can come in various guises). I have two questions: What is the scheme-theoretic automorphism group of the scheme $\texttt{Proj}(\mathbb{Z}[x, ...
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Decomposition of $\mathrm{Aut}(\mathbb{Z}_n)$

I know that $\mathrm{Aut}(\mathbb{Z}_n)\cong \mathbb{Z}_n^\times$ and in this thread a user correctly states that $\mathbb{Z}_n\cong \mathbb{Z}_{p_1}\times\dots\times \mathbb{Z}_{p_k}$, where the $p_i$...
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2 votes
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Degree of subfield fixed by single automorphism

Let $L/K$ be a finite Galois extension. If $\sigma \in \mathrm{Gal}(L/K)$ has order $d$, is it the case that $$L^\sigma := \{ \ell \in L : \sigma(\ell) = \ell\}$$ satisfies $[L^\sigma:K] = d$? This ...
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A subgroup $G$ of the group of analytic isomorphisms of the open unit disc $Aut(D(0,1))$ is the entire group

Suppose $G$ is a subgroup of $\text{Aut}(D(0,1))$ which contains all the origin -fixing rotations and at least one element other than these. Then the problem asks me to prove that $G$ is all of $\text{...
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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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Question regarding Galois group of a polynomial.

I am a graduate student.We have Galois theory in this semester.We were first taught splitting fields of a polynomial.Then our instructor introduced the Galois group of a polynomial $f\in F[x]$ to be $...
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How to find the automorphism of $\mathbb{Z_3} \times \mathbb{Z}_5$

We were told in class that the automorphism of $\mathbb{Z_3} \times \mathbb{Z}_5$ is congruent to $\mathbb{Z}_2 \times \mathbb{Z}_4$. I know that $Aut(G \times H) \cong Aut(G) \times Aut(H)$. However ...
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Classification of groups of order $18$

I was just going through the first classification- $$|G|=18=3^2\times 2$$ Then $G$ has a subgroup of order $9$(normal, say $K$) and a subgroup of order $2$ (say $H$). I want someone to help me with ...
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How do we find the homorphism from $\mathbb{Z_2} \to{\rm Aut}(\mathbb{Z_3} \times \mathbb{Z_3})$ [closed]

How do we find the homorphism from $\mathbb{Z_2} \to {\rm Aut}(\mathbb{Z_3} \times \mathbb{Z_3})?$ I know that ${\rm Aut}(\mathbb{Z_3} \times \mathbb{Z_3})$ is isomorphic to $GL_2(\mathbb{Z_3})$. We ...
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$p,q$ primes, $p\mid q-1$. Weaker assumption in the proof of the existence of non-trivial $C_p\ltimes C_q$?

Motivated by the fact that the non-existence of non-trivial $C_p\ltimes C_q$, for $p\nmid q-1$, can be proven without any piece of information on the structure of $\operatorname{Aut}(C_q)$, not even ...
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Difference between bijection, homeomorphism and autohomeomorphism?

What is the difference between an autohomeomorphism and a bijection? Suppose we work on topological spaces. Bijection = function between the elements of two sets, where each element of one set is ...
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1 vote
1 answer
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Order of automorphisms of non-abelian group

I'm trying to solve the problem, but it doesn't work. Please help me to solve it. Problem: prove that $\bigl| \operatorname{Aut} \, (G) \bigr| \ge 8$ if $G$ is non-abelian and not isomorphic with $S_3$...
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1 vote
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Automorphism and cyclicity

Let $G$ be a group. Show that if ${\rm Aut}(G)$ (the group of automorphism of $G$ ) is cyclic, then $G$ is abelian, and if $G$ is additionally finite, show that $G$ is cyclic. I need the second part ...
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Proof checking - Galois theory

I am trying to show that if $K$ is a field and $f \in K[x] $ has exactly $n$ distinct roots, say $\alpha_1 ,..., \alpha_n \in L $ where $L$ is a splitting field (so $L=K(\alpha_1 , ..., \alpha_n ) $) ...
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3 votes
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Finite abelian group and its automorphism

Let $G$ be a finite abelian group with $x, y \in G$ such that $|x| > 2$, $|y|$ divides $|x|$ and $y \notin \langle x \rangle$. Then there exists no automorphism $f$ on $G$ such that $f(x) = x$, $f(...
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Let $G$ be a non abelian simple group. Show that ${\rm Aut}(G^n)$ is isomorphic to ${\rm Aut}(G) \wr{\rm Sym}(n).$

I got this question but don't know how to answer it. Let $G$ be a non abelian simple group. Show that ${\rm Aut}(G^n)$ is isomorphic to ${\rm Aut}(G) \wr{\rm Sym}(n)$. I already know that ${\rm Aut}(G)...
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Intuition for finding a group $G$ such that $G \cong \mathrm{Aut}(G)$

I'm trying to find a group such that the map $G \to \mathrm{Aut}(G)$ sending $a$ to the the conjugation map $\phi_a (x) = axa^{-1}$ is an isomorphism. I know that $G = S_3$ works and I know how to ...
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Show $\mathrm{Aut}(G)$ is a group: Question on proving closure

This is a very narrow question I haven't been able to find answered elsewhere. Given a group $G$, I want to prove that $\mathrm{Aut}(G)$ is a group. All of the steps make sense to me, but I'm ...
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Proving there exists a category corresponding to a group (Clara Loeh pg-14)

Context: Definition 2.1.18 (Automorphism group). Let $C$ be a category and let $X$ be an object of $C$. Then the set ${\rm Aut}_C(X) $of all isomorphisms $ X \to X$ in $C$ is a group with respect to ...
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Is $\ker\varphi^2=(\ker\varphi)(\mathop{\rm im}\varphi\cap\ker\varphi^2)$ always true for group endomorphism $\varphi$?

Is $\ker\varphi^2=(\ker\varphi)(\mathop{\rm im}\varphi\cap\ker\varphi^2)$ always true for group endomorphism $\varphi$? It is trivial that $\ker \varphi^2 \supseteq (\ker \varphi) ( \mathop{\rm im}\...
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olympiad problem using the fact that $|G|\leq f(|\mathrm{Aut}(G)|)$

I've recently become aware of this result: Theorem 1. There exists a strictly increasing function $f:\mathbb{N}\to \mathbb{N}$, such that for any finite group $G$, the following inequality holds: $$|G|...
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How difficult is Gaussian elimination with automorphism?

Recently, I've had an exchange with someone about linearizing a randomly selected quasigroup. Let's say, this quasigroup is evaluated as $q(x,y) = Ax + By +c$ where $q$ is the quasigroup operation, $+$...
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G-equivariant automorphisms of G

Let $G$ be group, and $$ \operatorname{Aut}_G(G):=\{f\in\operatorname{Aut}G\mid f(xg)=f(x)g,\forall x,g\in G\} $$ I want to show that in fact $\operatorname{Aut}_G(G)\cong G$. $G\subset \operatorname{...
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Transitive Automorphism Groups of Steiner Systems

I recently did a project where I constructed the Mathieu groups $M_{11}$ and $M_{12}$ as automorphism groups from resp S(4,5,11) and S(5,6,12). I know that $M_{12}$ is sharply 5-transitive and $M_{11}$...
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Do the groups $\operatorname{SL}$, $\operatorname{PGL}$, and $\operatorname{PSL}$ over a field $K$ always have the same automorphism group?

Let $K$ be a field, then $\operatorname{GL}(n,K)$ consists of the $n\times n$ invertible matrices, $\operatorname{SL}(n,K)$ consists of the $n\times n-$matrices with determinant $1$, and $\...
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2 votes
1 answer
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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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2 votes
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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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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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Does there exist a group $G$ such that ${\rm Aut}(G) = S_6$?

I thought this would be an interesting question, since every other symmetric group on $n$ elements is possible as an automorphism group, since ${\rm Aut}(S_n) \cong S_n$ for $n \neq 2, 6$. Obviously, $...
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Proving that the group of holomorphic automorphisms of the Riemann Sphere $\mathbb{C}_\infty$ are the Möbius Transformations

Is the following proof correct? Proof: Let $F\in\text{Aut}(\mathbb{C}_\infty)$. Let $L:\mathbb{C}_\infty\to\mathbb{C}_\infty$ be a Möbius transformation that maps $F^{-1}(\infty)$ to $\infty$. For ...
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Concrete form of all finite order automorphisms of $\mathfrak{sl}_2$

A general automorphism of $\mathfrak{sl}_2$ is given by $x \mapsto \gamma x \gamma^{-1}$ where $x \in \mathfrak{sl}_2$ is written as a $2\times 2$ matrix and $\gamma \in \mathrm{SL}(2,\mathbb C)$ ...
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Is ${\rm Aut}(G)$ cyclic iff $G$ is cyclic of order $n = 1, 2, 4, p^k, 2p^k$ for any finite group $G$? [duplicate]

We know that ${\rm Aut}(C_n) \cong (\Bbb Z/n\Bbb Z)^*$. So, does this mean that for any finite group $G$ that ${\rm Aut}(G)$ is cyclic if and only if $G$ is cyclic of order $n = 1,2,4,p^k,2p^k$ since ...
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​automorphism group of rational numbers [duplicate]

Let $G=(\Bbb Q\setminus\{{0\}},∗)$ be a group of the rational numbers under multiplication. How to find the structure of ${\rm Aut}(G)$? I show that if $f$ is a automorphism then it maps a basis of $...
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Canonical graphs in nauty algorithm

I am trying to understand the nauty algorithm. I do not understand why the search tree is needed. As far as I understood after refining based on the neighbour information propagation a search tree is ...
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