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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What is a cogroup and what are coactions?

What is a cogroup and what are coactions? A very nice way to think about a group action on an object $X$ is as a group homomorphism from $G$ to $\operatorname{Aut}(X)$. Is there something similar for ...
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9 votes
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Who was the first to prove that the automorphism group of a finite field is cyclic and is generated by the Frobenius automorphism?

$\DeclareMathOperator\Aut{Aut}$It is well-known that the automorphism group $\Aut(F)$ of a finite field $F$ of characteristic $p$ is cyclic of order $n$ where $|F|=p^n$. Moreover, the cyclic group $\...
Taras Banakh's user avatar
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4 votes
2 answers
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Order of abelian subgroup of the automorphism group of an abelian group

Suppose $A$ is a finite abelian group and $B\leq\operatorname{Aut}(A)$ is abelian. Is it possible for the order of $B$ to be strictly greater than the order of $A$? What if I additionally impose that ...
tomasz's user avatar
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Is every pointwise-weakly continuous one-parameter group of automorphisms of B(H) given by a Hamiltonian?

Let $\mathcal H$ be a Hilbert space, $\mathscr B(\mathcal H)$ be the von Neumann algebra of all bounded operators on $\mathcal H$, and let $\sigma $ be a one-parameter group of automorphisms of $\...
Ruy's user avatar
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1 answer
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Finding automorphism groups of regular graphs [closed]

Can some body help me with some source code for finding automorphism groups of regular maps?. For example: the type of graph is denoted as $\{p, q\}$, which means that they are tessellations of the ...
Zahid Malik's user avatar
10 votes
0 answers
256 views

Looking for counterexamples: Are maximal tori in the automorphism groups of smooth complex quasiprojective varieties conjugate?

Let $X$ be a smooth quasiprojective variety over $\mathbb{C}$. It has a group of (algebraic) automorphisms $ \DeclareMathOperator{\Aut}{Aut} \Aut(X)$. Define a torus in $\Aut(X)$ to be a faithful ...
Carlos Esparza's user avatar
4 votes
0 answers
198 views

Infinite groups with 2 automorphism orbits

A group $G$ is called a $k$-orbit group if its automorphism group ${\rm Aut}(G)$ acting naturally on $G$ has precisely $k$ orbits. The only finite 2-orbit groups are the elementary abelian groups $(...
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Can a definable group of definable automorphisms of a field contain the Frobenius automorphism?

Let $K$ be an infinite definable field of characteristic $p >0$ in a certain theory $T$ with a definable group of definable automorphisms. Can this group contain the Frobenius automorphism?
Invictus's user avatar
2 votes
0 answers
124 views

Automorphisms of (nilpotent) groups : torsion cokernel on the abelianisation implies torsion cokernel on the center?

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\End{End}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Z{Z}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Span{Span}\DeclareMathOperator\ker{ker}\...
Christopher-Lloyd Simon's user avatar
5 votes
0 answers
150 views

Combinatorial classes where not almost all objects are asymmetric

Let $\mathcal{C} = \bigcup_{n=0}^{\infty}\mathcal{C}_n$ be a class of finite (labeled) combinatorial objects, where $\mathcal{C}_n$ is the set of objects on $[n] = \{1,2,\dotsc,n\}$. For example, $\...
Sam Hopkins's user avatar
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Determining homomorphism using automorphism group of two graphs

I wish to know the connection between the automorphism group of two graphs and homomorphism between them, if any. Like all Kneser graphs $K(n,k)$ have the same automorphism group $S_n$. But, given ...
vidyarthi's user avatar
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3 votes
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Are the automorphisms of the power semigroup of a cancellative semigroup cardinality-preserving?

Let $S$ be a semigroup (written multiplicatively) and $f$ be an automorphism of the power semigroup $\mathcal P(S)$ of $S$, that is, a bijective function on the family of all non-empty subsets of $S$ ...
Salvo Tringali's user avatar
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1 answer
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Holomorphic automorphisms of analytifications of proper varieties

I have two questions of a similar nature: Is it true that any holomorphic automorphism of an analytification of a proper variety over $\mathbb{C}$ is algebraic? If the first is not the case, is ...
Grisha Taroyan's user avatar
2 votes
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93 views

Automorphism group of the first Weyl field

A related question is this one (Automorphism group of the quantum Weyl field). Let $A_1$ denote the rank 1 Weyl algebra (over the complex numbers), and $D_1$ its skew field of fractions, called the ...
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5 votes
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Generalizations of Chevalley–Shephard–Todd's Theorem?

Major Edit I will reformulate my question signicantly, given Anton Geraschenko's comment. The old version of the question is bellow. For simplicity, my base field is $\mathbb{C}$. If $G<\...
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4 votes
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Automorphism group of a Lorentzian lattice

Consider the even integral lattices $L_n:=Z\times Z\times Z^{n-2}$ (where $Z$ is the set of integers) with elements $x=(x_+,x_-,x_d)$ and inner product $$(x,y):=x_+y_-+x_-y_++2x_d\cdot y_d.$$ Its ...
Arnold Neumaier's user avatar
3 votes
0 answers
95 views

Conjugate actions and isomorphic Zappa–Szép products

Let $A$ and $G$ be two cyclic groups. Let $\alpha:G\rightarrow\operatorname{Bij}(A)$ and $\beta: A\rightarrow\operatorname{Bij}(G)$ be two group homomorphisms satisfying some conditions given in ...
N. SNANOU's user avatar
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1 answer
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Lifting an automorphism of a curve to an automorphism of its Jacobian

Let $C: Y^3Z = f(X,Z)$, with $f(X,Z)\in K[X,Z]$, a degree 4 homogeneous polynomial, and $K$ a field. The curve $C$ has an order $3$ automorphism, given by sending $(x,y,z)$ to $(x,\omega y, z)$, where ...
kindasorta's user avatar
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Representability of twists of projective schemes

Let $K$ be a perfect field, and let $S$ be a projective $K$-scheme. Denote by $\text{Twist}(S/K)$ the set of twists of $S$ up to $K$-isomorphism. These are (apriori) sheaves $\mathcal{F}$ on the ...
kindasorta's user avatar
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Automorphism group of symmetric square

Say I have a hyperelliptic curve without any automorphism beyond the hyperelliptic involution. Is it possible for its symmetric square to obtain new automorphisms beyond the one induced by the ...
kindasorta's user avatar
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3 votes
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Galois action on automorphisms of a curve

Let $C$ be a smooth projective curve defined over a local field $K/\mathbb{Q}_p$. Denote by $\text{Aut}(C)$ the geometric automorphism group of $C$, which consist of isomorphisms of $C\times_{\text{...
kindasorta's user avatar
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3 votes
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Two more topologies on unitary groups

Let $H$ be a separable Hilbert space and let $\operatorname{U}(H)$ be the group of unitary transformations of $H$. It is well known that the weak, strong and compact-open topologies on $\operatorname{...
Matthias Ludewig's user avatar
4 votes
1 answer
235 views

If $\mathbb{C}(u(x,y),v(x,y),f(x))=\mathbb{C}(x,y)$, for every $f(x) \in \mathbb{C}[x]-\mathbb{C}$, then already $\mathbb{C}(u,v)=\mathbb{C}(x,y)$?

The following question is a direct continuation of this elaborate question; it is mentioned there at the end: Let $u,v \in \mathbb{C}(x,y)$ or $u,v \in \mathbb{C}[x,y]$, if it is easier to answer in ...
user237522's user avatar
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Representation of automorphism group of a curve acting on points of finite order in the Jacobian

Let $C$ be a curve of large genus $g > 1$ over an algebraically closed field of characteristic $0$, and let $G = \textrm{Aut}(C)$ be its automorphism group. Is there a general way to compute the ...
TCiur's user avatar
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Cocycle-conjugacy classes of flows on the C*-algebra of compact operators

A flow on a C*-algebra $A$ is a group homomorphism $\sigma $ from ${\mathbb R}$ into the group of *-automorphisms of $A$ such that the map $$ t\in {\mathbb R}\mapsto \sigma _t(a)\in A $$ is norm-...
Ruy's user avatar
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16 votes
1 answer
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A "simpler" description of the automorphism group of the lamplighter group

I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can point me to some relevant references. The lamplighter group is defined by the ...
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102 views

Automorphism group of conic bundle fixing the base

Let $\pi: X \to \mathbb{P}^n$ be a conic bundle over an (algebraically closed) field $k$. Let $g \in Aut(X)$ so that $g$ preserves the fibres of $\pi$. Clearly $g$ lives inside $PGL_3(k(\mathbb{P}^n))$...
TCiur's user avatar
  • 469
2 votes
1 answer
390 views

Algebraically closed fields with only finite orbits

The automorphism groups $\mathrm{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)$ of algebraic closures of finite fields $\mathbb{F}_q$ and the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\...
THC's user avatar
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2 votes
1 answer
66 views

Detecting non-affine automorphisms of a translation surface

Let $(X, \omega)$ be a translation surface, i.e., a Riemann surface with a homomorphism $1$-form. A central object is the group of affine automorphisms $\text{Aff}^+(X, \omega)$: homeomorphisms of $X$ ...
Sam Freedman's user avatar
8 votes
1 answer
308 views

Sylow $p$ of $\mathrm{Aut}(G)$ with $G$ finite simple?

I met the following problem: Let $G$ be a finite simple group (non-commutative, otherwise trivial). Let $p$ be a prime number not dividing $|G|$. Prove that any Sylow $p$ subgroup of $\mathrm{Aut}(G)$ ...
youknowwho's user avatar
2 votes
0 answers
112 views

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 ...
semisimpleton's user avatar
2 votes
0 answers
60 views

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 $...
THC's user avatar
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14 votes
2 answers
1k views

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 ...
A. Bailleul's user avatar
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2 votes
0 answers
78 views

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 $...
jg1896's user avatar
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6 votes
1 answer
478 views

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 ...
THC's user avatar
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3 votes
0 answers
119 views

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 ...
Leo Herr's user avatar
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15 votes
1 answer
700 views

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 ...
A. Bailleul's user avatar
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8 votes
1 answer
305 views

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 ...
Mike's user avatar
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1 vote
0 answers
115 views

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 $\operatorname{Hol}(Q_{2^{n+1}})$ ...
bidermeyer's user avatar
3 votes
1 answer
283 views

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)\...
user41650's user avatar
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8 votes
1 answer
408 views

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 ...
THC's user avatar
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7 votes
0 answers
162 views

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 ...
Rybin Dmitry's user avatar
3 votes
0 answers
35 views

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 ...
THC's user avatar
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4 votes
0 answers
100 views

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 ...
THC's user avatar
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8 votes
1 answer
244 views

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 ...
zjs's user avatar
  • 465
1 vote
1 answer
125 views

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 \...
THC's user avatar
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2 votes
1 answer
486 views

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]\...
1123581321's user avatar
6 votes
0 answers
96 views

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 ...
Matthias Ludewig's user avatar
3 votes
0 answers
140 views

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.
Fernando Muro's user avatar
1 vote
0 answers
33 views

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 $G$...
Sam Makhoul's user avatar

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