# 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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### How can I generate all unique row/column permutations of a given incidence (binary) matrix using group theory and matrix multiplication?

Given a binary matrix, we need to generate all distinct matrices that are isomorphic to this input matrix under row and column permutations. I think this problem might involve some group theory rather ...

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### Status of the automorphism tower problem for finite groups

This is problem 11.123 in the Kourovka notebook:
For a given group $G$, define the following sequence
of groups: $A_1(G) = G$, $A_{i+1}(G) = \operatorname{Aut}(A_i(G))$. Does there exist a finite ...

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### Countable highly order-transitive subgroups of $\mathrm{Aut}(\mathbb{Q},\leq)$

Consider $A := \mathrm{Aut}(\mathbb{Q},\leq)$, the group of order-automorphisms of $(\mathbb{Q},\leq)$. Call a subgroup $U$ highly order-transitive if for any two finite ordered sequences $s_1$ and $...

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### Is every group the automorphism group of a ring?

I know not all groups can be realized as the automorphism group of a group. For example, it is well-known that no group can have $\mathbb Z/n\mathbb Z$, with $n > 1$ odd, as automorphism group. Now ...

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### Automorphism group of the quantum Weyl field

Let $\mathsf{k}$ be a field with zero characteristic, and $q \in \mathsf{k}$ a non-zero elemento which is not a root of unit. The quantum plane $\mathsf{k}_q[x,y]$ is the algebra given by generators $...

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### Automorphisms of algebraically closed fields without the Axiom of Choice

In the paper Algebraische Konsequenzen des Determiniertheits-Axioms (U. Felgner – K. Schulz, Arch. Math. (Basel) 42 (1984), pp. 557–563), the authors show that in models of Zermelo-Fraenkel set theory ...

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### Finite algebras with finitely many automorphisms

Let $B'/B$ be a finite locally free algebra. Locally in $B$, there is an isomorphism of $B$-modules $B'\simeq B^{\oplus n}$. When is the automorphism group of $B'/B$ finite? When is it unramified? Is ...

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### Finite abelian groups with fewer automorphisms than a subgroup

It is not too hard to find examples of finite groups which have fewer automorphisms than one of their subgroups. For example $\mathcal D_4 \times \mathbb Z/2\mathbb Z$ (where $\mathcal D_4$ is the ...

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### If G is a finitely generated group with vcd(G) finite, is vcd(H) finite for H, where H is an automorphism group of G?

$\DeclareMathOperator\vcd{vcd}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Out{Out}$Here I mean $\vcd(G)$ to be the virtual cohomological dimension of $G$. Some ...

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### Is the commutator of the holomorph of generalized quaternion group abelian?

Let $Q_{2^{n}} = \langle x, y \mathrel\vert x^{2^{n-1}}=y^4 = 1, x^{2^{n-2}}=y^2, y^{-1}xy = x^{-1} \rangle$ be the generalized quaternion group of order $2^{n}$.
Let $\operatorname{Hol}(Q_{2^{n+1}})$ ...

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### Automorphism of moduli space of stable vector bundles over a curve

Let $C$ be a smooth genus two hyperelliptic curve and $\mathcal{M}_C$ be a moduli space of stable rank two vector bundles of fixed degree(or fixed determinant). Then is $\mathrm{Aut}(\mathcal{M}_C)\...

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### Automorphisms of projective spaces, and the Axiom of Choice

It is known that upon not accepting the Axiom of Choice (AC), there exist models of ZF in which there are projective spaces (over a division ring) with a trivial automorphism group. (This is a truly ...

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### Symmetrically $3d$ embeddable graphs with high girth

Let a symmetrically $3d$ embeddable graph be a graph that can be embedded into $3d$ so that the embedding is arc-transitive, which means that every vertex-edge pair with the vertex incident to the ...

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### Ring of invariants for graph automorphism

$\DeclareMathOperator\Aut{Aut}$Let $G$ be a finite simple graph with nodes numbered $1$ to $n$. Attach variables $x_1, ..., x_n$ to nodes. The graph automorphism group $\Aut G$ acts on nodes by ...

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### Baer involutions fixing the same plane

Let $\mathbf{PG}(2,q^2)$ be the finite projective plane defined over the finite field $\mathbb{F}_{q^2}$. Then for each quadrangle, there is precisely one involution fixing it pointwise, and hence ...

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### Two questions about $\mathbf{PSL}(V)$ with $V$ a vector space over a division ring

Let $V$ be a (possibly infinite-dimensional) vector space over a division ring $d$, and consider the projective special linear group $\mathbf{PSL}(V)$. We suppose that if the dimension of $V$ would be ...

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### Automorphisms of Lubotzky–Phillips–Sarnak graphs

For the Lubotzky–Phillips–Sarnak (LPS) graph $X^{p,q}$, what is its automorphism group? These graphs are not just ($p+1$)-regular but are Cayley graphs for $G=\mathrm{PSL}_2(\mathbb{F}_q)$, so clearly ...

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### Symmetry of infinite locally finite trees

Let $\Gamma$ be an infinite, locally finite tree without end points, with the additional property that there exists a positive integer $N$ such that for each occurring valency $v$, we have that $v \...

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### Growth rate of an outer automorphism of a free product

$\DeclareMathOperator\Out{Out}$Let $G=G_1\ast\cdots\ast G_k\ast F_p$ be a Grushko decomposition of a finitely generated group $G$, $\mathcal{O}$ the outer space relative to this decomposition, $[\phi]\...

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### Automorphisms of algebraic Clifford algebra of a Hilbert space

Let $H$ be a real separable, infinite-dimensional Hilbert space and let
$$\mathrm{Cl}(H) = \mathcal{T}(H_{\mathbb{C}}) / \{v\otimes w + w\otimes w - 2\langle v, w\rangle \cdot \mathbf{1} ~|~ v, w \in ...

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### Hochschild cohomology and outer automorphisms

Given an algebra $A$, I believe that the "conjugation" action of $\mathrm{Aut}(A)$ on $\mathrm{HH}^*(A)$ factors through $\mathrm{Out}(A)$. I’m looking for a reference.

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### Classifying endomorphisms of a direct sum Hilberts pace

Suppose I have a Hilbert space with a direct sum structure into "superselection sectors", i.e. $\mathcal{H} = \oplus_\alpha \mathcal{H}_\alpha$, where $\alpha$ labels irreps of some group $G$...

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### The geometry of the group of automorphisms of a manifold

Given a manifold $M$, the group $Aut(M)$ is made of diffeomorphisms $M\to M$. Since the complete vector fields on $M$ form an infinite dimensional Lie algebra, and each generates a 1 dimensional Lie ...

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### On cospectral graphs

Is there examples of non-isomorphic cospectral graphs having
Non-isomorphic automorphism groups?
Isomorphic automorphism groups?