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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8
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2answers
192 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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1answer
68 views

On cospectral graphs

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

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 $\...
13
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1answer
333 views

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 ...
7
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1answer
128 views

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. $\...
1
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1answer
96 views

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 ...
2
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1answer
175 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/...
8
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2answers
359 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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1answer
62 views

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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2answers
244 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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0answers
103 views

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 ...
7
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1answer
758 views

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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1answer
175 views

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 ...
5
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1answer
347 views

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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1answer
262 views

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 \...
5
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1answer
126 views

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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55 views

similar homomorphisms and conjugate subgroups

Suppose that $A$ is an abelian group and let $G$ be a finite group. Let $\alpha, \beta :G\rightarrow {\rm Aut}(A)$ be two injective homomorphisms. The homomorphisms $\alpha$ and $\beta$ are called ...
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337 views

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 ...
9
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1answer
503 views

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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0answers
142 views

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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1answer
92 views

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 ...
2
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1answer
281 views

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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303 views

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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110 views

Automorphism groups of Kähler–Einstein manifolds

Let $(X, \omega)$ be a compact Kähler manifold. We will say that $X$ is Calabi–Yau if the first Chern class of the anti-canonical bundle is trivial, in symbols: $c_1(-K_X)=0$; we will say $X$ is of ...
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1answer
229 views

Why $G/F(G)$ is isomorphic to a subgroup of ${\rm Out}(F(G))$?

I know two facts and I’ve managed to figure out how to prove one, but the other one is still a little confusing. Let $G$ be a finite solvable group and $F(G)$ is the Fitting subgroup of $G$. (1) $G/Z(...
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76 views

Action on cohomology by automorphisms of ihs manifolds

For all known deformation types of irreducible holomorphic symplectic manifolds (which I call K3, K3$^n$, Kum$^n$, OG$^3$, OG$^5$, the exponent being half the complex dimension), it is known that the ...
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0answers
147 views

On automorphism group

Let $p$ be a prime number, and let $\mathbb{F}_{p}$ be a finite field of order $p$. Let $U_{n}$ denote the unitriangular group of $n\times n$ upper triangular matrices with ones on the diagonal, over $...
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292 views

Which finite solvable groups have solvable automorphism groups?

Is it possible to give a reasonable description of those finite solvable groups $G$ such that $A = {\rm Aut}(G)$ is also solvable? The central case to deal with is that in which $G$ is a $p$-group of ...
12
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1answer
400 views

On the iterated automorphism groups of the cyclic groups

Let $C_n$ be the cyclic group of order $n$. Its automorphism group $Aut(C_n)$ is a group of order $\varphi(n)$ isomorphic to $(\mathbb{Z}/n\mathbb{Z})^{\times}$ the multiplicative group of integer ...
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0answers
34 views

Complexity of computing the automorphism group of the subdivision of clique with leaves

Related to graph isomorphism. Consider the graph transformation $G$ to $G'$. Make a clique of $V(G)$ and subdivide each edge once, i.e. replace edge $(u,v)$ with path $(u,S_{uv},v)$. For all edges $(...
10
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1answer
350 views

Small automorphism groups of groups

I do not know much about group theory, so sorry in case this question is not for MO. For a finite group $G$, denote by $f(G)$ the number of elements of the automorphism group of $G$. Question: For ...
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70 views

English translation of Fouxe-Rabinovitch paper

Is there somewhere an english translation of Fouxe-Rabinovitch's papers "D. I. Fouxe-Rabinovitch, Uber die Automorphismengruppen ¨ der freien Produkte. II, Rec. Math. [Mat. Sbornik] N.S., 1941, ...
5
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1answer
229 views

Can the automorphism group vary too much in families of complex projective varieties?

In a family of smooth projective curves over a reduced complex scheme of finite type the list of isomorphism classes of automorphism groups of the fibers is finite. This follows from the Hurwitz's ...
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0answers
109 views

Automorphism group of a hypersurface and its sections

This question is moved from my StackExchange. Assume the base field is algebraically closed. Let $X\subset \mathbb P^n$ be a fixed smooth hypersurface of degree $d$. For any hyperplane $H\subset \...
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0answers
51 views

Are there multiple conjugacy classes of order 2 elements in the smooth automorphism group of $\mathbb{R}$?

Consider the group $\text{Aut}\mathbb{R}$ of smooth invertible maps from $\mathbb{R}$ to $\mathbb{R}$. If $f\in\text{Aut}\mathbb{R}$ has order 2 ($f$ is an involution), is $f$ conjugate to $g(x)=-x$? ...
12
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2answers
395 views

Does asymmetric fraction of finite groups tend to $0$?

Let’s define asymmetric fraction of a finite group $G$ as the number $$\mathrm{af}(G) = \frac{|\{(g, a) \in G \times \mathrm{Aut}(G)\mid a(g) = g\}|}{|G|\cdot|\mathrm{Aut}(G)|}.$$ Equivalently it can ...
11
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1answer
312 views

On certain order-automorphisms of the rationals

Consider the rationals $\mathbb{Q}$ with the usual order $\leq$. Now let $A$ be a subset of $\mathbb{Q}$, such that foreseen with the induced order $\leq$, $(A,\leq)$ is a dense linear order. ...
2
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1answer
186 views

Fundamental theorem of linear orders

Let $(\Omega,\leq)$ be a countable linear order. Suppose that for every finite $m \in \mathbb{N}$, and all subsets $S_1$ and $S_2$ of $\Omega$ of order $m$, there is an order-automorphism of $(\Omega,\...
2
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0answers
56 views

What are the compact Aut(A) of an algebra A(G), G finite, that contains the identity?

If we have an algebra A over a finite group G, then if G is non-abelian we can have a non-trivial set of compact automorphisms of A that map the elements of G onto a set isomorphic to G. It may be ...
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0answers
103 views

Automorphism group of the cycle graph with $k$ diagonals

Let $C_n$ be the cycle graph on $n$ vertices. Then $Aut(C_n) \cong D_{2n}$ the dihedral group of order $n$. Now, let $C_{n,i}$ be the cycle graph with $i$ many diagonals connecting opposite vertices ...
4
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0answers
67 views

Are all verbal automorphisms inner power automorphisms?

Let $G$ be a group. $\DeclareMathOperator{\Wa}{Wa}\DeclareMathOperator{\Tame}{Tame}\DeclareMathOperator{\Aut}{Aut}$ Let's call $\phi \in \Aut(G)$ verbal automorphism iff $\exists n \in \mathbb{N}, \{...
3
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1answer
316 views

Infinite order automorphisms of planar polynomials

Let $R_n$ be the integral polynomial ring $\mathbb{Z}[x_1,x_2,...,x_n]$, let $A_n$ be the group of ring automorphisms $\mathrm{Aut}(R_n)$, and for $f\in R_n$ let $\mathrm{Aut}(f)=\{\alpha\in A_n\ |\ \...
2
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1answer
197 views

Abelian torsion-free group with $\mathbb{Z}_2\times\mathbb{Z}$ as automorphism group

Let $A$ be an abelian torsion-free group. If $A$ is isomorphic with the group of rational numbers whose denominators are powers of, say, $2$, then its automorphism group is isomorphic with $\mathbb{Z}...
6
votes
1answer
588 views

Automorphism group of the special unitary group $SU(N)$

Let us consider the automorphism group of the special unitary group $G=SU(N)$. We know there is an exact sequence: $$ 0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0. $$ For $G=SU(2)...
7
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0answers
209 views

Automorphism group of poset of number fields

Consider the poset of number fields, partial order being defined by inclusion of fields. What is the group of order-preserving automorphisms of this poset? What if we take only Galois extensions of $\...
4
votes
1answer
176 views

Metrics with prescribed Levi-Civita connection

My question involves the symmetries of a (pseudo)-Riemannian metric preserving the Levi-Civita connection (LCC), its unique torsion-free metric connection. For a basic example, one notes that the ...
1
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1answer
99 views

Sabidussi theorem for morphisms between graphs

Sabidussi proved that if a finite graph $X$ is isomorphic to a Cartesian product of connected graphs $X_1,\ldots,X_m$ which are pairwise relatively prime with respect to Cartesian multiplication, then ...
1
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1answer
523 views

What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in C^n ( n>1)? [closed]

What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in $\mathbb{C}^n$ ($n>1$) ? I would be pleased if you tell me.
9
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1answer
315 views

Automorphisms of $GL_n(\mathbb{Z})$

I want to consider the crossed module: $H \xrightarrow{t} Aut(H)$ for the case where $H = GL_n(\mathbb{Z}) = Aut(T^n)$ is the automorphism group of the $n$-torus. Any suggestions on how to understand ...

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