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.

learn more… | top users | synonyms

0
votes
2answers
42 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 ...
6
votes
2answers
337 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 ...
4
votes
2answers
215 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 ...
11
votes
1answer
457 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 ...
0
votes
0answers
197 views

Automorphisms of an L-function

Throughout this question, the term "L-function" will denote any element of the Selberg class. Following Strong automorphisms of the Selberg class, I define the group of automorphisms of an L-function ...
3
votes
2answers
316 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
2answers
231 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 ...
10
votes
1answer
308 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
0answers
101 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 ...
2
votes
0answers
67 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
1answer
217 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 ...
8
votes
2answers
518 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
347 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
4answers
314 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
0answers
219 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
1answer
147 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
2answers
387 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
1answer
350 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
1answer
135 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
1answer
436 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
0answers
84 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 $\varphi$ ...
2
votes
1answer
154 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 ...
0
votes
0answers
98 views

Automorphisms of the Galois group of a local field

Say $K$ is the field of fractions of a complete discrete valuation ring, and fix a separable closure of $K$. Is there an explicit description of the (continuous) outer automorphism of the absolute ...
13
votes
0answers
220 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
2answers
407 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 ...
3
votes
0answers
221 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 ...
4
votes
1answer
216 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
2answers
294 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
1answer
140 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
1answer
188 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
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 ...
0
votes
2answers
249 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
1answer
176 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
5answers
644 views
1
vote
0answers
173 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
1answer
282 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
1answer
233 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
1answer
199 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
4answers
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
1answer
293 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
0answers
205 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
2answers
883 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 ...
0
votes
1answer
232 views

Automorphisms in Hilbert spaces

Let $H$ be a Hilbert space and $H'\le H$ a subspace as Hilbert spaces (I mean, the inner product in $H'$ is the same inner product of $H$ restricted to $H'$). If we take $f:H\to H$ an automorphism of ...
1
vote
0answers
406 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 ...
6
votes
1answer
319 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).
4
votes
2answers
605 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 ...
5
votes
1answer
223 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 ...
3
votes
1answer
501 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 ...
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 ...
9
votes
2answers
459 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) ...