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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182 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
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2answers
531 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 ...
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5answers
813 views

Is every distance-regular graph vertex-transitive?

Is every distance-regular graph vertex-transitive?
7
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1answer
337 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
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1answer
257 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!
6
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2answers
377 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 ...
2
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1answer
216 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 ...
3
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1answer
323 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
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0answers
209 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 ...
11
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2answers
1k 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
252 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 ...
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0answers
427 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
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1answer
337 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
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1answer
237 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
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1answer
550 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
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1answer
2k 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 ...
4
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2answers
659 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 ...
9
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2answers
475 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) ...
2
votes
1answer
494 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 ...
5
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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
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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 ...
4
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2answers
470 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 ...
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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 ...