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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22
votes
2answers
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 ...
14
votes
0answers
228 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 ...
11
votes
2answers
1k views

Realizing groups as automorphism groups of graphs.

Frucht showed that every finite group is the automorphism group of a finite graph. The paper is here. The argument basically is that a group is the automorphism group of its (colored) Cayley graph ...
11
votes
1answer
477 views

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

I become interested in this problem because $G\cong Aut(G)$ suggests a special symmetry in $G$. From $G/Z(G)\cong Inn(G)$ we know complete group is the anewer for the simplest case, though this class ...
11
votes
1answer
331 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 $G$ ...
11
votes
1answer
359 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 ...
9
votes
2answers
463 views

Outer automorphisms of free groups into bigger free groups

This may be very either very simple or very unknown, but here goes: Let $F_n$ be the free group on $n$ generators and $Out(F_n)$ its outer automorphism group. Can embeddings $Out(F_n) ...
9
votes
1answer
246 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 ...
9
votes
2answers
888 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 ...
8
votes
2answers
429 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 ...
7
votes
4answers
353 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 ...
7
votes
1answer
305 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 ...
7
votes
1answer
1k views

Sylow subgroups invariant under an automorphism

Let $G$ be a finite group and $\sigma$ an automorphism of $G$. Suppose $p$ is a prime and $\sigma$ has prime order $q \neq p$. It's easy to see that $\sigma$ fixes a Sylow $p$-subgroup of $G$ if ...
7
votes
2answers
522 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 ...
6
votes
4answers
317 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 ...
6
votes
1answer
450 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 = ...
6
votes
1answer
324 views

Automorphism groups of virtually cyclic groups

Let $V$ be a virtually cyclic group. Then is $Aut(V)$ also a virtually cyclic group? This is true when $V$ is a finite group (zero-ended) and when $V = C_\infty, D_\infty$ (both two-ended).
6
votes
1answer
231 views

A restatement, in terms of the semi-group product of the left-invariant completion of a Polish group, of http://mathoverflow.net/questions/71389

This is a re-statement, of sorts, of Is there a relational countable ultra-homogeneous structure whose countable substructures do not have the amalgamation property?, so far unanswered. Let $G$ be a ...
6
votes
2answers
373 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 ...
5
votes
3answers
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 ...
4
votes
2answers
621 views

Automorphism Group of some Classical groups

Hi All, I would like to know the Automorphism group of some simple classical groups, such as PSL(n,q) or some PSU or PSp groups. Could you please give me some recommended books or papers then? I ...
4
votes
2answers
438 views

Groups as automorphism groups of small graphs and the number of rigid graphs of a given size

In a recent question of mine I asked whether every infinite group is (isomorphic to) the automorphism group of a graph. The finite case was done by Frucht in 1939. The first answer to this ...
4
votes
1answer
219 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 $F_2$ ...
4
votes
2answers
226 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 ...
4
votes
1answer
1k views

Non-trivial consequences of Baer's theorem and Lucchini's theorem in subnormality theory

There are a couple of beautiful results in finite group theory that look trivial, at least on a first glance, but require non-trivial facts to prove. I am basically interested in whether these results ...
3
votes
5answers
715 views

Is every distance-regular graph vertex-transitive?

Is every distance-regular graph vertex-transitive?
3
votes
2answers
331 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 ...
3
votes
2answers
400 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 ...
3
votes
2answers
224 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 ...
3
votes
2answers
325 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? ...
3
votes
1answer
513 views

Automorphisms of an infinite direct product of abelian groups

Let $G = \prod_p \mathbb{Z}/p\mathbb{Z}$, where $p$ ranges over all primes, considered as an abelian group. What does $\text{Aut}(G)$ (or even $\text{End}(G)$) look like? I know that that if we take ...
3
votes
1answer
301 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
1answer
240 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!
3
votes
3answers
215 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 ...
3
votes
1answer
149 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
0answers
80 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 ...
3
votes
0answers
261 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 ...
3
votes
0answers
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 ...
2
votes
1answer
203 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 ...
2
votes
2answers
135 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 ...
2
votes
1answer
470 views

Automorphism of a wreath product

Let $S_k \wr S_n$ be the wreath product of two symmetric groups (so $S_n$ acts on $(S_k)^n = S_k \times ... \times S_k$ by permuting the factors; we then take the semi-direct product). What is ...
2
votes
1answer
164 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 ...
2
votes
1answer
140 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 ...
2
votes
0answers
237 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 ...
2
votes
0answers
90 views

Groups of automorphisms of weighted graphs

Let $\Gamma=(V,E,\omega)$ be an (edge-)weighted graph without loops and multiple edges. Here $V$ is the set of vertices, $E$ is the set of edges and $\omega:E \to \mathbb{N}$. A permutation $\varphi$ ...
1
vote
2answers
237 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 ...
1
vote
1answer
192 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 ...
1
vote
0answers
104 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 ...
1
vote
0answers
178 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 ...
1
vote
0answers
412 views

Equivalence classes induced on binary strings by set of permutations

Let $\mathbb{F}_2^{n}$ be the set of binary strings of length $n$ and let $f: \mathbb{F}_2^{n} \rightarrow \mathbb{R}$ be a function from the set of binary strings of length $n$ to the reals. Let's ...