**12**

votes

**1**answer

677 views

### Find finite groups $G\cong Aut(G)$

I become interested in this problem because $G\cong Aut(G)$ suggests a special symmetry in $G$ (This kind of group describes its own symmetry).
From $G/Z(G)\cong Inn(G)$ we know complete group is the ...

**2**

votes

**1**answer

135 views

### Graph automorphism that swaps two pairs of nodes

Suppose we have two automorphisms on a graph $G$ such that each one swaps a separate pairs of vertices. Is it possible to construct (or prove the existence of) a third automorphism that swaps both ...

**13**

votes

**6**answers

706 views

### Does the linear automorphism group determine the vector space?

I was recently thinking about what it means to put structure on a set. It seems to me that, in my area (representation theory), the two main ways of imposing structure on a set $X$ are:
...

**6**

votes

**1**answer

536 views

### Automorphisms of Generic Abelian Varieties

Automorphism groups of elliptic curves are very well understood. Of course, every elliptic curve has the automorphism $[-1]$ of order $2$. If we are over a (algebraically closed) field, this is the ...

**2**

votes

**1**answer

151 views

### When is Aut(G) the symmetric group of an Aut(G)-invariant generating set?

Let $G$ be a group, $X$ a generating set of $G$. Suppose $X$ is $\operatorname{Aut}(G)$-invariant, i.e. $\sigma(X)\subseteq X$ for all $\sigma \in \operatorname{Aut}(G)$. When is the restriction ...

**0**

votes

**1**answer

54 views

### edge transitivity and edge deletion

Let G be a graph which has the following properties:
1) For every $e_1,e_2 \notin E(G)$, $G \cup e_1 \cong G \cup e_2$
2) For every $e_1,e_2 \in E(G)$, $G\setminus e_1 \cong G\setminus e_2$
i.e. ...

**3**

votes

**1**answer

138 views

### Automorphism group of a smooth quadric $Q\subset\mathbb{P}^4$

Let $Q$ be the smooth quadric threefold in $\mathbb{P}^4_{\mathbb{C}}$ defined by the equation
$x_0x_4+x_1x_3+x_2^2=0$.
Is it true that the automorphism group of $Q$ is $SO(Q;\mathbb{C})$ which is ...

**4**

votes

**1**answer

139 views

### Examples of hyperelliptic curves with hyperelliptic quotients that have more automorphisms

Does there exist a hyperelliptic curve $X$ of genus $g\geq 2$ over the complex numbers such that $X$ has a hyperelliptic quotient $X\to Y$ (in the sense that $Y$ is hyperelliptic and the morphism ...

**4**

votes

**1**answer

180 views

### Automorphisms of SO_n(k,f)

Let $k$ be a field, $n\in\mathbb{N}$ and $f:k^n\times k^n\to k$ a non-degenerate symmetric bilinear form. Let
$$O_n(k,f):=\{ g\in GL_n(k) \mid \forall x,y\in k^n : f(x,y)=f(g.x,g.y) \}$$
and
...

**4**

votes

**3**answers

258 views

### Is the group of isometries of a homogeneous Riemannian manifold maximal?

I have a homogeneous Riemannian manifold X with isometry group Iso. Is Iso a maximal group? By maximal group, I mean that there does not exist another group G such that:
Iso is a proper subgroup of ...

**6**

votes

**2**answers

374 views

### How big $|Aut(M)|$ can be, given $|\partial Aut(M)|$?

My apologies: There were a couple of typos in the original question. Hope I got them all.
Let $\kappa$ be an uncountable cardinal of cofinality $\omega$ and $M$ a model of size $\kappa$. We equip ...

**0**

votes

**1**answer

124 views

### Automorphism group of an affine curve

Let $q$ be a power of an odd prime.
Consider the affine curve $\mathcal C$ defined over $\mathbb F_q$ by $y^2=\prod_{\xi\in\mathbb F_q}(x-\xi)$. I try to determinate the $\mathbb F_q$-automorphism ...

**2**

votes

**2**answers

140 views

### $PSL_2(\mathbb{Z}/p^n)$ isomorphic to automorphism group of depth-$n$, $(p+1)$-regular tree?

A comment on another question (linked below) states
"The group $PSL_2((\mathbb{Z}/p^n))$ is the automorphisms group of the $(p+1)$ regular tree of depth $n$, where at level $m$ of the tree you have ...

**3**

votes

**2**answers

260 views

### The group of diffeomorphisms with compact support

Let $M$ be a topological/differentiable manifold. Is there any topology on the group of homeomorphisms/diffeomorphisms with compact support, turning it into a (locally-)compact topological group?
(My ...

**0**

votes

**2**answers

54 views

### Lie Automorphisms and Isotopy

Let $X$ be a Lie group, $Aut(X)$ be the Lie automorphism group of $X$ (group automorphisms which are also diffeomorphisms), and $Homeo(X)$ be the homeomorphism group of the underlying manifold. For ...

**4**

votes

**2**answers

250 views

### Root system automorphisms as inner automorphisms of extended Chevalley group

For each automorphism $\sigma$ of a root system $\Phi$ there is a unique automorphism of the Chevalley group $G(\Phi,R)$ such that $\sigma(x_\alpha(t))=x_{\sigma\alpha}(t')$. While conjugating by ...

**3**

votes

**2**answers

374 views

### A structure of the group of automorphisms of an infinite binary tree

My friend asked me to ask his question here. Where he can find (a paper or a book) containing a complete description (with the proof) of a structure of the group of automorphisms of an infinite binary ...

**1**

vote

**2**answers

240 views

### Elementary group theory question

Is the following statement true in general?
If $G$ is any group and $K(G)=\{x \in G ; \alpha(x)=x , \forall \alpha \in Aut_c(G)\}$ when $Aut_c(G)$ is the subgroup of all central automorphisms. Then ...

**11**

votes

**1**answer

358 views

### Iterated Automorphism Groups

Notation: For each group $G$ define:
$Aut^{(0)}(G):=G$
$Aut^{(1)}(G):=Aut(G)$
$\forall n\geq 1~~~Aut^{(n+1)}(G):=Aut(Aut^{(n)}(G))$
Question: Consider $I\subseteq \omega$. Is there a group $G$ ...

**1**

vote

**0**answers

112 views

### Automorphism on F_2[[X,S]]

Let us define the automorphism $\sigma$ on ${\Bbb F}_2[[X,S]]$ such that
$\sigma \colon S \mapsto S + S^2 + S^3$
$\sigma \colon X \mapsto X + S$.
It is easy to see that the ideal $(S)$ is stable ...

**3**

votes

**0**answers

92 views

### Certain subgroup of automorphism groups of binary codes

Suppose that $C$ is an binary linear code of length $n$ and dimension $k$ (i.e. it's a $k$-dimensional linear subspace of $\mathbb{F}_2^n$). As usual, the automorphism group of $C$ is the subgroup of ...

**9**

votes

**1**answer

278 views

### Do varieties with ample canonical bundle have finite automorphism group in small characteristic?

Suppose $X$ is a smooth projective variety over a field $k$, with ample canonical bundle. If $\operatorname{char}(k)=0$ or $\operatorname{char}(k)>\dim(X)$ and $X$ lifts to $W_2(k)$ (thanks ...

**7**

votes

**2**answers

529 views

### Automorphisms of subgroup of hamming cube under distance constraint

Let $S$ be a subset of $\{0,1\}^n$ such that any two elements of $S$ are at least (Hamming) distance 5 apart. I'm looking for an upper bound on the size of the automorphism group of $S$.
There's a ...

**7**

votes

**4**answers

368 views

### On the fixed point of automorphism of F_3[[T]]

Consider the automorphism $\sigma$ on ${\Bbb F}_3[[T]]$ such that $T \mapsto c_1T + f(T)$ with $c_1 = 1$ or $-1$, and $f(T) \not=0$ and the non-zero leading term $c_mT^m$ of $f(T)$ satisfies $m \geq ...

**6**

votes

**4**answers

327 views

### Is the conjugacy problem solvable in $Out(F_n)$?

There is a paper of Martin Lustig on his webpage giving a positive answer to the conjugacy problem for the outer automorphism group of the free group $F_n$. On the other hand, there seems not to be a ...

**2**

votes

**0**answers

247 views

### Automorphisms of the L-function associated to an elliptic $\mathbb{Q}$-curve

Edited after Noam Elkies' comment: From what I understand (very few actually), there exist elliptic curves defined over some number fields $\mathbb{K}$ Galois over $\mathbb{Q}$ which are isogenous to ...

**3**

votes

**1**answer

151 views

### What are the endomorphisms of the group of affine transformations of a field?

Let $k$ be a field (I am mostly concerned with $k=\mathbb{Q}$) and let $A$ denote the group of affine transformations of $k$. In other words $A$ is (isomorphic to) the group
...

**3**

votes

**2**answers

434 views

### Monotonic bijections of rational numbers

How can one characterize monotonic bijections from $\mathbb{Q}$ to
$\mathbb{Q}$? It is easy to see that piecewise linear functions which are
strictly monotonic and surjective will do the trick, but ...

**11**

votes

**1**answer

381 views

### Automorphisms of $P(\Bbb N)$

I believe I've proved that the power semigroup of non-negative integers with addition has a trivial automorphism group. The proof is a bit long, completely elementary and rather unremarkable (as the ...

**2**

votes

**1**answer

145 views

### $Aut(\mathbb{Z}G)=?$ for $G=\mathbb{Z}^2\rtimes_n\mathbb{Z}$

I am interested in the automorphism group of the group ring $\mathbb{Z}G$ for some noncommutative group $G$ of the form $\mathbb{Z}^2\rtimes_n\mathbb{Z}$, say
...

**6**

votes

**1**answer

479 views

### Automorphism group of a finite group

I would like to ask if there exists an explicit description of $\mathrm{Aut}(G)$, the group of automorphisms of a finite group $G$, in particular, when $G$ is abelian. E.g., if $G = ...

**2**

votes

**1**answer

171 views

### Fixed points of IA automorphisms

Let $F_n$ denote the free group on $n$ generators $x_1,\ldots , x_n$.
Recall that an element $\varphi\in\mathrm{Aut}(F_n)$ is an IA automorphism if it induces the identity on the abelianization ...

**15**

votes

**0**answers

264 views

### Maximum automorphism group for a 3-connected cubic graph

The following arose as a side issue in a project on graph reconstruction.
Problem: Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a ...

**8**

votes

**2**answers

449 views

### Proving that a generic variety with ample canonical bundle has no automorphisms

Let $X$ be a smooth projective connected variety over the complex numbers with ample canonical bundle. If $X$ is generic and $\dim X \leq1$, the automorphism group of $X$ is trivial, see for instance
...

**4**

votes

**1**answer

226 views

### Automorphism classes of the free group

As is well known, the conjugacy classes of the free group $F_2$ are parametrised by cyclically reduced words, up to cyclic permutation. In particular, it's easy to tell whether two elements of $F_2$ ...

**3**

votes

**2**answers

393 views

### Automorphism group of a compact Kahler manifold

Good evening,
I would like to ask the following questions.
Let $X$ be a compact Kahler manifold. Denote by Aut(X) the group of all the biholomorphisms of $X.$
1) What can we say about this group? ...

**0**

votes

**1**answer

155 views

### Affine automorphisms of algebraic function field towers

Are there any well-known towers of function fields over finite fields whose automorphism groups contain a transitive subgroup consisting solely of affine maps?
For a (non)example of what I'm looking ...

**0**

votes

**1**answer

217 views

### When is a cyclic cover hyperelliptic?

Let us work over the complex numbers for simplicity. Consider a curve $C$ presented as a cyclic cover of some lower genus curve $C'$. When $C'$ has genus $0$, we can write $C$ as the normalization of ...

**22**

votes

**2**answers

1k views

### Does this poset have a unique minimal element?

Recently I have been thinking about the following poset: the underlying set is $\mathcal{AFT}$ consisting of all (finite) automorphism-free undirected trees (with at least one edge to exclude the ...

**0**

votes

**2**answers

302 views

### Confused about orbits

I am trying to apply the main theorem of this paper to a certain kind of graph and keep getting confused. The theorem uses $rank(Aut\Gamma)$ which is defined as "the number of $Aut \Gamma$ orbits on ...

**1**

vote

**1**answer

215 views

### On linear automorphism on positive definite matrices.

I saw a statement in [Murakami, On automorphisms on Siegel domains] that every linear automorphism $\phi$ on the set of positive definite matrices can be represented as conjugation: i.e. there is a ...

**3**

votes

**5**answers

774 views

### Is every distance-regular graph vertex-transitive?

Is every distance-regular graph vertex-transitive?

**1**

vote

**0**answers

180 views

### Outer automorphisms of an infinite simple group

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition
$\tau_{r_1(m_1),r_2(m_2)}$ be the ...

**7**

votes

**1**answer

328 views

### Maximal subgroups of a certain finite 2-group

The following came up in a problem on reconstruction of digraphs. I determined enough about the answer to satisfy the application completely, but still I am curious to know what the complete solution ...

**3**

votes

**1**answer

250 views

### Eigenvectors of asymmetric graphs

Let $G$ be an asymmetric connected graph. Then is it always the case that at least one of the eigenvectors of its adjacency matrix $A$ consists entirely of distinct entries?
Thanks!

**2**

votes

**1**answer

213 views

### Countable structures with uncountable many automorphisms

The following is supposed to be "clear" according to Kueker, but I could not see why. Can anyone help?
Let $A$ be a countable structure with uncountable many automorphisms. Then for every $\vec{a}\in ...

**5**

votes

**3**answers

1k views

### The automorphism group of a hyperelliptic curve

Let $C$ be the projective smooth genus 2 curve defined by $y^2=x^5-x$ over $\mathbb F_5.$ What is the order of its automorphism group (automorphisms over $\mathbb F_5$)?
I have seen different ...

**3**

votes

**1**answer

314 views

### Automorphism group of factor groups

Let $G$ be a group and let $H$ be a factor group of $G$. Is there any result that relates $\operatorname{Aut}(G)$ (the automorphism group of $G$) and $\operatorname{Aut}(H)$?
As a very special case ...

**3**

votes

**0**answers

208 views

### When Aut(M) preserves a linear order?

I have a general-type question:
Suppose $M$ is a countable structure that is ultrahomogeneous, i.e. every (partial) isomorphism between finitely generated substructures of $M$ extends to an ...

**9**

votes

**2**answers

896 views

### Example of a Group which has $\text{SL}_{n}(\mathbb{Z})$ as the automorphism group

For the past one week, I have been trying to learn more about Automorphism groups of different groups. Very recently one of my friend asked this question to me:
What is the Autmorphism group of ...