# 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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### 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 ...

**3**

votes

**0**answers

74 views

### Reference of generalized isometries

I'm wondering if these objects have a name or are studied. No one around me knows, so I thought to ask here.
Let $\Phi:\mathbb{R}^d\rightarrow \mathbb{R}^d$ be $C^2$-diffeomoprhism, and fix $p \geq ...

**11**

votes

**1**answer

386 views

### Is there a name of semidirect product of a group with its automorphism group?

Consider the construction $G \rtimes \text{Aut}(G)$. Here $
G$ is a group, $\text{Aut}(G)$ is the automorphism group and the semidirect product is over the most obvious action.
1) Is there any name ...

**3**

votes

**1**answer

165 views

### How large can a symmetric generating set of a finite group be?

Let $G$ be a finite group of order $n$ and let $\Delta$ be its generating set. I'll say that $\Delta$ generates $G$ symmetrically if for every permutation $\pi$ of $\Delta$ there exists $f:G\...

**0**

votes

**0**answers

74 views

### Functoriality of indiscernible sequences

Let $T$ be a first order theory of, say, some type of combinatorial geometries which contain indiscernible sequences of points. Let $(\Gamma,\mathcal{O})$ be a model of $T$, where $\Gamma$ is the ...

**13**

votes

**1**answer

276 views

### 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-...

**4**

votes

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133 views

### Groups inducing edge-colorings on graphs. Is this concept known?

Are the following concepts known in graph/group theory, and if Yes, what are they called and where to read about them? Because I do not know better, I gave them placeholder names for now.
1. ...

**9**

votes

**2**answers

371 views

### Reference request: birational automorphism group is finite

I am interested in having a look at the proof of the following fact: If $X$ is a smooth variety of general type, then $\mathrm{Aut(X)}$ is finite.
I know that this is proved in "On algebraic groups ...

**3**

votes

**1**answer

75 views

### A partition of the set of order 2 outer automorphisms of $SU(N)$

Let $N$ be an even integer, $N>2$. Let $E$ be the set of all outer automorphisms $\phi$ of $G = SU(N)$ which are of order 2, i.e. $\phi \circ \phi = \mathrm{id}_G$.
Choose a particular element $\...

**7**

votes

**2**answers

283 views

### What are the automorphism groups of direct products of dihedral group D4

What is the automorphism group of direct sum of dihedral group of order $8$, $D_4$?
For example, $\mathrm{Aut}(D_4)$ is isomorphic to $D_4$. How about $\mathrm{Aut}(D_4\times D_4)$, $\mathrm{Aut}(...

**2**

votes

**1**answer

253 views

### Schreier conjecture — without a simple proof? and sporadic simple groups

The Schreier conjecture asserts that $\mathrm{Out}(G)$ is always a solvable group when $G$ is a finite simple group. This result is known to be true as a corollary of the classification of finite ...

**10**

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377 views

### On the Number of Parallel Automorphism Lines

Given a group $G$, one can define the transfinite line of iterative automorphisms of $G$ to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the ...

**11**

votes

**1**answer

842 views

### The Tall Tale of Terminating Transfinite Towers

The transfinite tower of iterative automorphisms of a group $G$ is simply definied to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the direct ...

**1**

vote

**1**answer

123 views

### Lie brackets of automorphisms

Let $F$ be the vector fields of a differential manifold $M$, let $[X,Y]$ be the Lie brackets of $F$, now let $a$ be an automorphism of $F$ for the structure of real vector space of $F$. I consider now ...

**11**

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251 views

### An example of curves with the same Jacobian, but different Jacobian automorphism groups (wrt their respective canonical principal polarizations)?

I am trying to understand examples of differing curves with the same Jacobian, and the quirks of the Jacobian.
Here is my question: What is an example where $Aut(Jac(X, a))$ and $Aut(Jac(X', a'))$ ...

**3**

votes

**1**answer

78 views

### automorphisms of a measurable space can be approximated by continuous measure preserving maps?

Suppose first that $D=[0,1]$ is equipped with the usual Lebesgue measure, and that $\varphi$ is a measure-preserving transformation $\varphi:D\to D$ that is bijective and whose inverse is also measure ...

**5**

votes

**1**answer

233 views

### $\operatorname{Out}(F_n)$ is not linear for $n > 3$

The paper The Tits alternative for $\operatorname{Out}(F_n)$ I by Bestvina, Feighn and Handel and the paper Automorphisms of free groups and Outer space by Vogtmann both state that $\operatorname{Aut}(...

**5**

votes

**0**answers

102 views

### Sign preserving Galois automorphisms

I have an algebraic number $\alpha \in \mathbb{Q}(\zeta)$, where $\zeta^n = 1$ is a root of unity (not primitive) given as a linear combination of powers of $\zeta$, i.e, $\alpha = \sum_{i=1}^k a_i \...

**9**

votes

**1**answer

365 views

### The Tits alternative for $\operatorname{Out}(F_n)$

Not sure if this is the right place to ask this, but the paper I am reading seems to be too specialised for mathstack (if you do not agree, pleas let me know and I will take down this question)
I am ...

**4**

votes

**2**answers

164 views

### How do eigenvalues of combinatorial Laplacian relates to automorphisms in graphs?

Is there a relation between eigenvalues of the graph Laplacian and the automorphism group of a simple graph?
How are the multiplicities of Laplacian eigenvalues related to the order of the ...

**12**

votes

**2**answers

572 views

### Graph automorphism group

Let $A_w$ denote such set of positive integer $n$ that: for any two permutations $\pi_0,\pi_1\in S_n$, if $\pi_1$ is not a power of $\pi_0$, then there exists a (labeled non oriented) graph $G$ of ...

**9**

votes

**2**answers

225 views

### How to characterize “matching-transitive” regular graphs?

I am interested in regular graphs $G$ such that for each pair of 1-factors (=perfect matchings) $F$ and $F'$ there is an automorphism of $G$ that takes $F$ to $F'$. Let's call this property matching ...

**7**

votes

**3**answers

350 views

### When do automorphisms on open subsets extend

Let $X$ be a normal affine variety of dimension at least two over $\mathbb{C}$ and let $U\subset X$ be a dense open. Assume that $\mathrm{codim}(X\setminus U) \geq 2$.
I think Hartog's lemma implies ...

**18**

votes

**3**answers

528 views

### If a variety $X$ has finite automorphism group, is the same true for its $n$-fold self-products?

Let $X$ be an algebraic variety over $\mathbb{C}$.
Let $n\geq 1$ be an integer and let $X^n$ be the $n$-fold self product of $X$.
Q. Is there an integer $n\geq 1$ and an algebraic variety $X$ ...

**13**

votes

**4**answers

411 views

### Non-split Aut(G) $\to$ Out(G)?

I'm looking for examples of outer automorphisms of a finite group $G$ which do not lift to automorphisms (i.e. non-split quotient map $\mathrm{Aut}(G)\to \mathrm{Out}(G)$, where $\mathrm{Out}(G) = \...

**4**

votes

**0**answers

175 views

### Is finiteness of automorphism groups a birational question?

Let $X$ be a proper variety over $\mathbb{C}$ and let $Y\to X$ be a proper birational surjective morphism.
Is $Aut(X)$ finite if and only if $Aut(Y)$ is finite?
The answer is no. Indeed, $\mathbb{P}^...

**2**

votes

**2**answers

177 views

### question on simple groups [closed]

Let $G$ be a simple finite group. Then every non trivial endomorphism of $G$ is an automorphism. My question is: does the converse holds? More precisely, if $G$ is a finite group all of whose non ...

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votes

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89 views

### Automorphisms of the modular product of graphs

Given graphs $X$ and $Y$, their modular product, which I will denote by $X \diamond Y$, has vertex set $V(X) \times V(Y)$ where two vertices $(x,y)$ and $(x',y')$ are adjacent if ($xx' \in E(X)$ and $...

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vote

**1**answer

147 views

### Automorphism group of fiber products of schemes

Let $A \mapsto S$ and $B \mapsto S$ be two schemes over the scheme $S$. Is there a connection between the automorphism group of the scheme $A \otimes_{S} B$ and the automorphism groups of $A$ and $B$ ?...

**15**

votes

**1**answer

296 views

### Birational automorphisms of varieties of Picard number one

Let $X$ be a smooth projective variety of Picard number one, and let $f:X\dashrightarrow X$ be a birational automorphism which is not an automorphism.
Must $f$ necessarily contract a divisor?

**1**

vote

**0**answers

55 views

### Automorphism groups of graphs of bounded treewidth

The celebrated Frucht's theorem states that every finite group is isomorphic to the automorphism group of a finite graph $G$. If we restrict $G$ to belong to a certain class, some groups may become ...

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votes

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79 views

### Automorphisms of rationally connected varieties

Let $X$ be a smooth, rationally connected variety over an algebraically closed field of characteristic zero. Denote by $\mathrm{Aut}(X)$ the space of automorphisms of $X$ and for a given $\phi \in \...

**5**

votes

**1**answer

151 views

### Permutations of points in the projective plane

Let $p_1,...,p_7\in\mathbb{P}^{2}$ be seven general points in the projective plane $\mathbb{P}^{2}$ over the complex numbers.
Let $f$ be an automorphism of $\mathbb{P}^{2}$ inducing a permutation of $...

**1**

vote

**1**answer

174 views

### Automorphism group of a graph

Suppose $\Gamma$ is a simple graph and $G=\mathrm{Aut}(\Gamma)$ is the automorphism group of $\Gamma$. If $G$ stabilizes a subgraph $\Gamma_1$,, and $G_0$ is the point-wise stabiliser of the set $V(\...

**6**

votes

**1**answer

162 views

### Automorphism group of $UT(n,p)$, the group of unitriangular matrices over the field $\mathbb{F}_p$

I am interested in understanding (at least roughly, if no such a description exists) the group of automorphisms for the group $UT(n,p)$, of unitriangular matrices over the field $\mathbb{F}_p$ on $p$-...

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199 views

### On Kalai's $3^{d}$ conjecture

I just learned the existence of Gil Kalai's $3^{d}$ conjecture, which according to Wikipedia, is proven for $d$ at most $4$. It states that every $d$ dimensional polytope with central symmetry has at ...

**21**

votes

**1**answer

564 views

### Finite-order self-homeomorphisms of $\mathbf{R}^n$

Consider the $n$-dimensional euclidean space $\mathbf{R}^n$. A self-homeomorphism $\phi:\mathbf{R}^n\to \mathbf{R}^n$ is said to be of finite order if $\phi^m = \mathrm{id}_{\mathbf{R}^n}$ for some ...

**2**

votes

**0**answers

89 views

### On the determinant of incidence matrices (of graphs and other geometries)

Let $\Gamma = (P,L,I)$ be a point-line geometry (here, $P$ is the point set, $L$ the line set, and $I$ is the symmetric incidence relation). (As an example, $\Gamma$ could be a graph.) I suppose $\...

**1**

vote

**1**answer

57 views

### Uniqueness of polytope embedding from symmetry group

Do the symmetry group generators of a regular convex polytope and a marked $\{0,1\}^n$ vertex point suffice to embed the polytope uniquely with $\{0,1\}^n$ vertex set?
If so can we find the John's ...

**3**

votes

**1**answer

182 views

### Approximating John's ellipsoid from uniform sampling of a centrally symmetric convex polyhedron

A centrally symmetric convex polyhedron in $\Bbb R^n$ shifted from the origin with possibly $e^{\alpha n}$ number of vertices at some $\alpha>0$ has an unique ellipsoid of maximum volume called ...

**6**

votes

**1**answer

391 views

### “Jacobian Conjecture” for $k[x_1,\ldots,x_n,x_1^{-1},\ldots,x_n^{-1}]$?

Is there exist a similar conjecture to the famous Jacobian Conjecture with $\mathbb{C}[x_1,\ldots,x_n,x_1^{-1},\ldots,x_n^{-1}]$ instead of
$\mathbb{C}[x_1,\ldots,x_n]$?
Namely, let $f$ be $\mathbb{...

**1**

vote

**0**answers

119 views

### Involutions on $\mathbb{C}(x,y)$

How to find all involutions on $\mathbb{C}(x,y)$,
or at least all involutions $\delta$ on $\mathbb{C}(x,y)$ such that
$\delta(x)=x$?
Remarks:
(1) An involution on $\mathbb{C}[x,y]$ is either ...

**2**

votes

**1**answer

260 views

### Compact dual of a noncompact Lie group

Let $\mathfrak{g}_0$ be a noncompact simple Lie algebra, and fix a Cartan involution $\theta$ of $\mathfrak{g}_0$, which gives a Cartan decomposition $\mathfrak{g}_0=\mathfrak{k}_0+\mathfrak{p}_0$. ...

**3**

votes

**0**answers

87 views

### The group of automorphisms and anti-automorphisms of the first Weyl algebra

Let $k$ be a field of characteristic zero, and let $A_1=A_1(k)$ be the first Weyl algebra.
It is well known (first proved by Dixmier, if I am not wrong) that the group of automorphisms of $A_1$, ...

**11**

votes

**1**answer

535 views

### Sheaf associated to presheaf Aut

Let $S$ be a scheme and let $C$ be the category of schemes flat and locally of finite presentation over $S$. Endow $C$ with the fppf topology (or perhaps any subcanonical topology). Let $\mathcal P$ ...

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vote

**0**answers

97 views

### automorphism group of tensor product of two graph

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 ...

**2**

votes

**0**answers

395 views

### 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 ...

**6**

votes

**1**answer

195 views

### Conditions for a finite group to be isomorphic to its automorphism group

So in the interest of gaining a better understanding of a conjecture (due to Scott, 1960) on the automorphism series (first part of the automorphism tower, no direct limits) of a finite group that ...

**4**

votes

**1**answer

196 views

### Extremal rays of the effective cone

Let $X$ be a smooth projective variety with polyhedral finitely generated effective cone $Eff(X)$. Let $f:X\dashrightarrow X$ be a birational automorphism of $X$ that is an isomorphism in codimension ...

**5**

votes

**1**answer

436 views

### What is the maximal order of the automorphism group of a given Shimura variety?

Background:
Given an elliptic curve $E$, it seems that $max(ord(Aut(E)))$ over the prime 2 is 24, and $(max(ord(Aut(E)))$ over the prime 3 is 12.
The endomorphism algebra of an elliptic curve over $...