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.
294
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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
$$A=\left\{\left(\begin{...
- 1,701
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2
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925
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Root system automorphisms as inner automorphisms of extended Chevalley group
For each automorphism $\sigma$ of a root system $\Phi$ there is a unique automorphism of the Chevalley group $G(\Phi,R)$ such that $\sigma(x_\alpha(t))=x_{\sigma\alpha}(t')$. While conjugating by ...
- 3,101
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1
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$Aut(\mathbb{Z}G)=?$ for $G=\mathbb{Z}^2\rtimes_n\mathbb{Z}$
I am interested in the automorphism group of the group ring $\mathbb{Z}G$ for some noncommutative group $G$ of the form $\mathbb{Z}^2\rtimes_n\mathbb{Z}$, say
$$\mathbb{Z}^2\rtimes_n\mathbb{Z}=\...
- 1,528
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votes
1
answer
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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 ...
- 1,435
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1
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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 = \mathbb{Z}/m\...
- 655
2
votes
1
answer
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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 $F_n/[...
- 35k
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0
answers
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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 ...
- 37.2k
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2
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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
...
- 236
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1
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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 ...
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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 $F_2$ ...
- 9,195
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2
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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? ...
- 758
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votes
1
answer
210
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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 ...
- 53
0
votes
1
answer
426
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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 ...
- 243
4
votes
2
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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 ...
- 155
24
votes
2
answers
1k
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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 ...
- 5,181
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2
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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 ...
- 6,882
2
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1
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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 ...
- 537
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0
answers
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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 ...
- 19.5k
7
votes
2
answers
609
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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 ...
- 71
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Is every distance-regular graph vertex-transitive?
Is every distance-regular graph vertex-transitive?
- 151
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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 ...
- 37.2k
3
votes
1
answer
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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!
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2
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How big $|\operatorname{Aut}(M)|$ can be, given $|\partial\operatorname{Aut}(M)|$?
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 $\...
- 2,932
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votes
1
answer
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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 ...
- 2,932
4
votes
1
answer
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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 ...
- 1,108
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0
answers
301
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When does $\operatorname{Aut}(M)$ preserve a linear order?
I have a general-type question:
Let $M$ be a countable structure that is ultrahomogeneous, i.e. every (partial) isomorphism between finitely generated substructures of $M$ extends to an automorphism ...
- 2,932
15
votes
2
answers
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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 automorphism group of $(\...
- 4,735
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votes
2
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Example of an infinite abelian but non-cyclic group whose automorphism group is cyclic
Can anyone give me an example of:
An infinite abelian but non-cyclic group whose automorphism group is cyclic.
- 4,735
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votes
1
answer
582
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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 ...
- 3
2
votes
0
answers
775
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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 ...
- 73
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votes
1
answer
425
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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).
- 677
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Semigroup product of the left-invariant completion of a Polish group (restatement of Question 71389)
This is a re-statement, of sorts, of the question Is there a relational countable ultra-homogeneous structure whose countable substructures do not have the amalgamation property?, so far unanswered.
...
- 587
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votes
1
answer
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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 $...
- 3,001
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1
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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 $\...
- 97
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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 ...
- 233
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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) \...
- 1,160
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1
answer
761
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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 $Aut(...
- 1,160
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votes
3
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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 ...
- 4,195
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votes
1
answer
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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 ...
- 1,973
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2
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679
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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 ...
- 16.4k
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2
answers
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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
...
- 16.4k
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 ...
- 61.5k
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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))....
- 12.7k
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4
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Is ${\rm S}_6$ the automorphism group of a group?
The automorphism group of the symmetric group $S_n$ is $S_n$ when $n$ is not $2$ or $6$, in which cases it is respectively $1$ and the semidirect product of $S_6$ with the (cyclic) group of order $2$. ...
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