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.
293
questions
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Automorphism group of tensor product of two graphs
Is there any relation between the automorphism group of the tensor product of two graphs $G = G_1 \times G_2$ and the automorphism groups of $G_1$ and $G_2$?
I am aware of the nice results for the ...
4
votes
2
answers
214
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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 ...
9
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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 $\...
9
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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 ...
4
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0
answers
122
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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 $\...
0
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1
answer
100
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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 ...
4
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0
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193
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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 $(...
14
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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 ...
14
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2
answers
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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 ...
2
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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}\...
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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?
5
votes
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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, $\...
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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 ...
3
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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 $S$ ...
3
votes
2
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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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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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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 ...
2
votes
0
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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 ...
5
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The group of automorphisms of a polynomial ring in two variables over an integral domain
It is well-known that the group of automorphisms of a polynomial ring
$k[x,y]$, $k$ is any field, is a free product of $A(2)$ and $J(2)$
amalgamated along their intersection,
where $A(2)$ is its ...
4
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0
answers
200
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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 ...
3
votes
0
answers
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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 ...
0
votes
1
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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 ...
1
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0
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121
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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 ...
36
votes
5
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When is Aut(G) abelian?
Let $G$ be a group such that $\operatorname{Aut}(G)$ is abelian. Is then $G$ abelian?
This is a sort of generalization of the well-known exercise, that $G$ is abelian when $\operatorname{Aut}(G)$ is ...
0
votes
0
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84
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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 ...
3
votes
1
answer
196
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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{...
3
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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{...
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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 ...
4
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1
answer
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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 ...
8
votes
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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 $\mathrm{Aut}(G)$ ...
1
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0
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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-...
16
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1
answer
791
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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 ...
14
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1
answer
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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.
...
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0
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98
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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 $PGL_3(k(\mathbb{P}^n))$...
2
votes
1
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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}}/\...
2
votes
1
answer
66
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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 $X$ ...
2
votes
0
answers
111
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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 ...
2
votes
0
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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 $...
8
votes
1
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240
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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 ...
171
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7
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Does $\DeclareMathOperator\Aut{Aut}\Aut(\Aut(\dots\Aut(G)\dots))$ stabilize?
Purely for fun, I was playing around with iteratively applying $\DeclareMathOperator{\Aut}{Aut}\Aut$ to a group $G$; that is, studying groups of the form
$$ {\Aut}^n(G):= \Aut(\Aut(\dots\Aut(G)\dots))....
2
votes
0
answers
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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 $...
6
votes
1
answer
477
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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 ...
3
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0
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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 ...
15
votes
1
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697
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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 ...
8
votes
1
answer
305
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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 ...
18
votes
2
answers
557
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Can the graph of a symmetric polytope have more symmetries than the polytope itself?
I consider convex polytopes $P\subseteq\Bbb R^d$ (convex hull of finitely many points) which are arc-transitive, i.e. where the automorphism group acts transitively on the 1-flags (incident vertex-...
1
vote
0
answers
113
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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 $\operatorname{Hol}(Q_{2^{n+1}})$ ...
6
votes
1
answer
371
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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 ...
3
votes
1
answer
280
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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)\...
8
votes
1
answer
406
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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 ...