Questions tagged [automorphism-groups]

Questions about the group of automorphisms of any mathematical object $X$ endowed with a given structure, i.e the group of all bijective maps from $X$ to itself preserving this structure, and hence helping study it further and understand it better.

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Nonlinear automorphisms of projective spaces and the axiom of choice

Let $k$ be a field and $\mathbf{P}$ a projective space over $k$. If we accept the axiom of choice (AC), then $\mathbf{P}$ has a basis and a dimension $m$, and if $m$ is finite, the automorphism group ...
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What is mean of action of group on tower?

What is mean of action of group on tower ?in the page 17 of following paper (action on teichmuller tower ) In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class ...
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Automorphism group of a K3-surface

I am interested to know more about the automorphism group of a K3 surface, more specifically: is there any easy way to determine if it is infinite or not? The specific case I am looking at is the ...
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What is "inn" in this paper?

In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030.,in the page 18 we have: $$ \begin{aligned} ...
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Anti-flag transitive affine planes

Let $\mathcal{A}$ be an axiomatic affine plane. First let $\mathcal{A}$ be finite. Suppose that the automorphism group of $\mathcal{A}$ acts transitively on nonincident point-line pairs (that is, on ...
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Proving that the fundamental group of a finite Galois category is profinite

This is sort of a cross-posting of this question of mine over on math.SE. Suppose that I am given a finite Galois category $(\mathcal{G}, F)$, i.e. a Galois category in the sense of SGA 1. One of the ...
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Is a finite order automorphism of k[x,y] necessarily linear?

Let $k[x,y]$ be the polynomial ring in two variables over a field $k$ of characteristic zero. Every $k$-algebra automorphism of $k[x,y]$ is tame (e.g. the paper of McKay and Wang). It was pointed out ...
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170 views

Automorphisms of symmetric powers of projective space

$\newcommand{\pt}{\mathit{pt}}$For $d>1$ is it possible to understand $\text{Aut}(\text{Sym}^n(\mathbb{P}^d_{\mathbb{C}}))$? Automorphisms mean biregular morphisms from the variety to itself. Not ...
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Is there a natural topology on the automorphism group of a topological group?

$\DeclareMathOperator\TAut{TAut}\DeclareMathOperator\Homeo{Homeo}$Let $G$ be a topological group, and let $\TAut(G)$ denote the group of topological automorphisms of $G$ under composition (i.e. the ...
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Is there a name for objects all of whose endomorphisms are automorphisms?

I am looking for a descriptive adjective to describe the following special property that some objects in some categories enjoy: their endomorphism monoids are groups. Of course, one way this could ...
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Poset of automorphism groups of variants of a structure

Say that two structures $\mathfrak{A},\mathfrak{B}$ in possibly different finite languages are parametrically equivalent - and write "$\mathfrak{A}\approx\mathfrak{B}$" - iff they have the ...
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Automorphism groups of symmetric graphs

Which groups can occur as automorphism groups of arc-transitive graphs?
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Automorphism group of indefinite orthogonal Lie group $G=O(p,q)$ vs that of a double covering group $\tilde{G}$

Previously I mentioned in Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same or not? that the automorphism group of a Lie group 𝐺 may be the same as that of ...
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Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same or not? [duplicate]

Previously there are some counterexamples given in Automorphism group of a Lie group $G$ vs that of a covering group $\tilde G$: same or not? such that the automorphism group of a Lie group 𝐺 is not ...
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Automorphism group of a Lie group $G$ vs that of a covering group $\tilde G$: same or not?

Is it true or false that the Inner (Inn), Outer (Out) and Total (Aut) Automorphism of a Lie group $G$ is the same as the covering group of the Lie group, say $\tilde G$ (regardless of how many types ...
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Automorphisms of the ring of periods

The set of periods $\mathcal{P}$ introduced by Kontsevich and Zagier forms a ring, see for example https://en.m.wikipedia.org/wiki/Period_(algebraic_geometry). Moreover J. Wan introduced in 2011 in ...
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Groups $G$ such that $\mathrm{Aut}(G) \simeq \mathbb{Z}/2\mathbb{Z}$

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Inn{Inn}$First I hope my question belongs here, please let me know if it doesn't. It isn't too hard to show there is no groups $G$ such that $\Aut(G) ...
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Automorphism groups of simple groups of Lie type

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PGL{PGL}$In “Automorphisms of finite linear groups”, Steinberg proves that any automorphism of a simple group of Lie type (normal or twisted) is a ...
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Outer and inner automorphism of $\mathrm{Pin}$ groups

$\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator\Pin{Pin}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSO{...
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4 votes
1 answer
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Do almost-point-transitive algebras generate almost-point-transitive varieties?

Say that an algebra $\mathfrak{A}$ (in the sense of universal algebra) is point-transitive iff for every $a,b\in\mathfrak{A}$ there is a $\pi\in Aut(\mathfrak{A})$ with $\pi(a)=b$. While genuinely ...
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7 votes
1 answer
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Is $\mathbb{Q}$ "equivalent" to a structure with transitive automorphism group action?

Say that structures $\mathfrak{A},\mathfrak{B}$ with the same underlying set are parametrically equivalent iff every primitive relation/function in one is definable (with parameters) in the other. For ...
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-3 votes
1 answer
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Structure of the automorphism group of an L-rig

This question is a follow-up to Are there infinitely many L-rigs? which is already pretty convoluted. Define the $\varphi$-evaluation morphism at a complex number $s$ as $\epsilon_{\varphi,s}:F\mapsto ...
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A question on Galois groups and deformations of Galois coverings

I've recently been working on several problems related to deformations of Galois covers of projective space, and have come across the following situation: If $X$ is a projective variety and $G$ is a (...
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Automorphism groups of which lattices act irreducibly on the ambient Euclidean space

(I asked this question on MSE a few days ago but it hasn't drawn any response yet.) Let $V$ be a finite-dimensional real inner product space and let $L \subset V$ be a lattice of full rank. Consider ...
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2 votes
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Johnson filtration and lower central series

Let $G$ be a group and consider the lower central series: $$G=\gamma_1 G \geq \gamma_2 G=[G,\gamma_1 G]\geq \gamma_3G=[G,\gamma_2G]\geq\cdots.$$ Let $S_g^1$ be a compact oriented genus $g$ surface ...
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1 vote
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Automorphisms of the topological field $\mathbb{C}_p$ of $p$-adic complex numbers?

I am interested to see what is currently known about the automorphisms of the topological field $\mathbb{C}_p$ of $p$-adic complex numbers (with respect to the $p$-adic topology induced by the $p$-...
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1 vote
1 answer
54 views

inequivalent vertex weights on finite poset

Let $m\geq1$ and $P$ be an arbitrary poset with vertex set $V=\{v_1,\dots,v_n\}$, edge set $E,$ and set $O$ of orbits under $\text{Aut}(P).$ Can we efficiently generate all inequivalent nonnegative ...
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8 votes
2 answers
237 views

Is the automorphism group of free group of rank two relatively hyperbolic?

By Behrstock, Drutu and Mosher [BDM], we know that the (outer) automorphism groups $\mathrm{Aut}(F_n)$ and $\mathrm{Out}(F_n)$ of free group of rank $n$ are not relatively hyperbolic if $n \geq 3$ (...
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Aut/Inn/Out Automorphism Groups of the unitary group $𝑈(𝑁)$

Given a group $G$, we denote the center Z$(G)$, we like to know the automorphism group Aut($G$), the outer automorphism Out($G$) and the inner automorphism Inn($G$). They form short exact sequences: $$...
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3 votes
1 answer
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On cospectral graphs

Is there examples of non-isomorphic cospectral graphs having Non-isomorphic automorphism groups? Isomorphic automorphism groups?
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5 votes
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Copies of the reals in $\mathbb{C}$ without the Axiom of Choice

Suppose we work in a model in which the Axiom of Choice does not hold, and in which $\mathbb{C}$ only has one nontrivial automorphism (such models exist). Question: "how many" subfields of $\...
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14 votes
1 answer
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Why is the automorphism group of the octonions $G_2$ instead of $SO(7)$

I can calculate the derivation of the octonions and I clearly find the 14 generators that form the algebra of $G_2$. However, when I do the same calculation for the quaternions, I end up with the ...
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7 votes
1 answer
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Howson property of automorphism group of $F_2$ and of $F_3$

Is the intersection of any two finitely generated subgroups of $\operatorname{Aut}(F_2)$ (resp. $\operatorname{Aut}(F_3)$) again finitely generated? That is, does $\operatorname{Aut}(F_2)$ (resp. $\...
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1 vote
1 answer
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Automorphisms of the ring of Laurent polynomials

Is the group of automorphisms of the ring $\mathbb{F}[t,t^{-1}]$ of Laurent polynomials known? Here, $\mathbb{F}$ is an algebraically closed field of characteristic $0$ and I am considering not ...
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2 votes
1 answer
183 views

Automorphisms of $G/Z(G)$ with $G$ simply connected

Let $G$ be a simply connected (if necessary, compact Lie) group with finite center $Z$ and $p:G/\to G/Z$ be the canonical projection. Is there any way to know if every element in $\operatorname{Out}(G/...
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8 votes
2 answers
412 views

On $p$-groups with abelian automorphism group

Let $G$ be a $p$-group of order $p^{n}\geq p^{7}$ and its automorphism group is elementary abelian $p$-group. Then, it is clear that $G$ is nilpotent of class $2$. However, the converse is not true in ...
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1 vote
1 answer
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When is a infinite transcendence-degree rigid fields fixed by a finite extension?

A field is rigid if it has no nontrivial automorphisms. Let $F$ be a rigid field which has infinite transcendence degree over $\mathbb{Q}$, and let $E$ be a finite extension of $F$. Then my question ...
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4 votes
2 answers
263 views

Can every involution of a symmetric directed graph be written as a power of another symmetry?

Let $D=(V,A)$ be a finite directed graph, and suppose that $D$ is vertex-transitive, $D$ is edge-transitive, and between any two vertices there is at most one edge, in particular, if $(v,w)\in A$ ...
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1 vote
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Is it always possible to improve the derivation by exponential conjugations?

$\DeclareMathOperator\Sp{Sp}$Let $R = k(t)[x_1,\ldots, x_n]$ be the polynomial algebra over the field $k(t)$ (where $k$ is assumed to be algebraically closed). Also let $Q = k[t,x_1,\ldots, x_n]$. Fix ...
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8 votes
1 answer
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Automorphisms over finite field that do not lift to an automorphism in characteristic zero

My main question is the following: is there an automorphism of the affine space $\mathbb{A}^n$ (automorphism of an algebraic variety) defined over a finite field which does not lift to an automorphism ...
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1 vote
1 answer
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An automorphism of a function field

I am looking for an explicit example of a function field other than rational with an automorphism which fixes places of different degrees. Also, is there a counterexample, namely a function field with ...
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6 votes
1 answer
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What are the "simplest" polytopes with an automorphism group of $\mathrm M_{11} \hspace {-1.25pt} $?

Do any polytopes have an automorphism group of the smallest of the sporadic groups, the Matthieu group $\mathrm M_{11} \hspace {-1pt} $? Indeed, they must exist. What are the simplest such polytopes ...
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Does Aut(G) → Out(G) always split for a compact, connected Lie group G?

The outer automorphism group of a topological group $G$ is constructed by the short exact sequence $$ 1\longrightarrow \operatorname{Inn}(G) \longrightarrow \operatorname{Aut}(G) \longrightarrow \...
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5 votes
1 answer
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Outer automorphism group of posets

Let $X$ be a finite poset (we can assume it is connected) and $A_K(X)$ the incidence algebra of $X$ over a field $K$. The following result is well known, see for example corollary 7.3.7 in the book &...
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7 votes
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Automorphism groups of the complex numbers, and other fields

If one accepts the Axiom of Choice (AC), then the automorphism group of $\mathbb{C}$ is a huge and wild group, very poorly understood. But apparently if one does not accept the Axiom of Choice, then ...
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11 votes
1 answer
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What is the automorphism group of the projective line minus $n$ points?

$\DeclareMathOperator{\AGL}{\operatorname{AGL}}\DeclareMathOperator{\PGL}{\operatorname{PGL}}$What is the automorphism group of $\mathbb P^1$ minus $n$ points (let's say over an algebraically closed ...
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2 votes
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Automorphisms group of complex and real simple Lie algebras

$\DeclareMathOperator{\Inn}{\operatorname{Inn}}\DeclareMathOperator{\Aut}{\operatorname{Aut}}\DeclareMathOperator{\Out}{\operatorname{Out}}\DeclareMathOperator{\g}{\mathfrak{g}}$According to Wikipedia,...
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3 votes
1 answer
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Spectral properties of half-transitive graphs

The half-transitive graphs form a curious class of graphs with some kind of intermediate symmetry that is non-trivial to achieve. More precisely, a graph is half-transitive if its symmetry group is ...
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2 votes
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Explicit automorphism map of ${\rm Spin}(8;\mathbb{R})$, ${\rm SO}(8;\mathbb{R})$, ${\rm PSO}(8;\mathbb{R})$

$\DeclareMathOperator{\SO}{\mathrm{SO}}\DeclareMathOperator{\Spin}{\mathrm{Spin}}\DeclareMathOperator{\Inn}{\mathrm{Inn}}\DeclareMathOperator{\Out}{\mathrm{Out}}\DeclareMathOperator{\Aut}{\mathrm{Aut}}...
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10 votes
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Does the tropicalization of a curve remember the curve's automorphism group?

For a tropical curve $Z$, let us call $Z_0$ this curve with its 1-valent points removed. (Def [5] of Joyner-Ksir-Grant Melles) Let the automorphism group of a tropical curve $Z$ be a map $g: Z \to Z$ ...
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