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.

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On the iterated automorphism groups of the cyclic groups

Let $C_n$ be the cyclic group of order $n$. Its automorphism group $Aut(C_n)$ is a group of order $\varphi(n)$ isomorphic to $(\mathbb{Z}/n\mathbb{Z})^{\times}$ the multiplicative group of integer ...
Sebastien Palcoux's user avatar
2 votes
0 answers
38 views

Complexity of computing the automorphism group of the subdivision of clique with leaves

Related to graph isomorphism. Consider the graph transformation $G$ to $G'$. Make a clique of $V(G)$ and subdivide each edge once, i.e. replace edge $(u,v)$ with path $(u,S_{uv},v)$. For all edges $(...
joro's user avatar
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10 votes
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391 views

Small automorphism groups of groups

I do not know much about group theory, so sorry in case this question is not for MO. For a finite group $G$, denote by $f(G)$ the number of elements of the automorphism group of $G$. Question: For ...
Mare's user avatar
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2 votes
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114 views

English translation of Fouxe-Rabinovitch paper

Is there somewhere an english translation of Fouxe-Rabinovitch's papers "D. I. Fouxe-Rabinovitch, Uber die Automorphismengruppen ¨ der freien Produkte. II, Rec. Math. [Mat. Sbornik] N.S., 1941, ...
1123581321's user avatar
5 votes
1 answer
295 views

Can the automorphism group vary too much in families of complex projective varieties?

In a family of smooth projective curves over a reduced complex scheme of finite type the list of isomorphism classes of automorphism groups of the fibers is finite. This follows from the Hurwitz's ...
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3 votes
0 answers
154 views

Automorphism group of a hypersurface and its sections

This question is moved from my StackExchange. Assume the base field is algebraically closed. Let $X\subset \mathbb P^n$ be a fixed smooth hypersurface of degree $d$. For any hyperplane $H\subset \...
Akatsuki's user avatar
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Are there multiple conjugacy classes of order 2 elements in the smooth automorphism group of $\mathbb{R}$?

Consider the group $\text{Aut}\mathbb{R}$ of smooth invertible maps from $\mathbb{R}$ to $\mathbb{R}$. If $f\in\text{Aut}\mathbb{R}$ has order 2 ($f$ is an involution), is $f$ conjugate to $g(x)=-x$? ...
Anon E. Mous's user avatar
12 votes
2 answers
406 views

Does asymmetric fraction of finite groups tend to $0$?

Let’s define asymmetric fraction of a finite group $G$ as the number $$\mathrm{af}(G) = \frac{|\{(g, a) \in G \times \mathrm{Aut}(G)\mid a(g) = g\}|}{|G|\cdot|\mathrm{Aut}(G)|}.$$ Equivalently it can ...
Chain Markov's user avatar
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14 votes
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On certain order-automorphisms of the rationals

Consider the rationals $\mathbb{Q}$ with the usual order $\leq$. Now let $A$ be a subset of $\mathbb{Q}$, such that foreseen with the induced order $\leq$, $(A,\leq)$ is a dense linear order. ...
THC's user avatar
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2 votes
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Fundamental theorem of linear orders

Let $(\Omega,\leq)$ be a countable linear order. Suppose that for every finite $m \in \mathbb{N}$, and all subsets $S_1$ and $S_2$ of $\Omega$ of order $m$, there is an order-automorphism of $(\Omega,\...
THC's user avatar
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What are the compact Aut(A) of an algebra A(G), G finite, that contains the identity?

If we have an algebra A over a finite group G, then if G is non-abelian we can have a non-trivial set of compact automorphisms of A that map the elements of G onto a set isomorphic to G. It may be ...
James Bellinger's user avatar
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0 answers
208 views

Automorphism group of the cycle graph with $k$ diagonals

Let $C_n$ be the cycle graph on $n$ vertices. Then $Aut(C_n) \cong D_{2n}$ the dihedral group of order $n$. Now, let $C_{n,i}$ be the cycle graph with $i$ many diagonals connecting opposite vertices ...
GA316's user avatar
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Are all verbal automorphisms inner power automorphisms?

Let $G$ be a group. $\DeclareMathOperator{\Wa}{Wa}\DeclareMathOperator{\Tame}{Tame}\DeclareMathOperator{\Aut}{Aut}$ Let's call $\phi \in \Aut(G)$ verbal automorphism iff $\exists n \in \mathbb{N}, \{...
Chain Markov's user avatar
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3 votes
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Infinite order automorphisms of planar polynomials

Let $R_n$ be the integral polynomial ring $\mathbb{Z}[x_1,x_2,...,x_n]$, let $A_n$ be the group of ring automorphisms $\mathrm{Aut}(R_n)$, and for $f\in R_n$ let $\mathrm{Aut}(f)=\{\alpha\in A_n\ |\ \...
Sean Lawton's user avatar
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Abelian torsion-free group with $\mathbb{Z}_2\times\mathbb{Z}$ as automorphism group

Let $A$ be an abelian torsion-free group. If $A$ is isomorphic with the group of rational numbers whose denominators are powers of, say, $2$, then its automorphism group is isomorphic with $\mathbb{Z}...
Alex Doe's user avatar
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7 votes
1 answer
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Automorphism group of the special unitary group $SU(N)$

Let us consider the automorphism group of the special unitary group $G=SU(N)$. We know there is an exact sequence: $$ 0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0. $$ For $G=SU(2)...
annie marie cœur's user avatar
7 votes
0 answers
223 views

Automorphism group of poset of number fields

Consider the poset of number fields, partial order being defined by inclusion of fields. What is the group of order-preserving automorphisms of this poset? What if we take only Galois extensions of $\...
user138266's user avatar
4 votes
1 answer
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Metrics with prescribed Levi-Civita connection

My question involves the symmetries of a (pseudo)-Riemannian metric preserving the Levi-Civita connection (LCC), its unique torsion-free metric connection. For a basic example, one notes that the ...
pre-kidney's user avatar
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Sabidussi theorem for morphisms between graphs

Sabidussi proved that if a finite graph $X$ is isomorphic to a Cartesian product of connected graphs $X_1,\ldots,X_m$ which are pairwise relatively prime with respect to Cartesian multiplication, then ...
THC's user avatar
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1 answer
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What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in C^n ( n>1)? [closed]

What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in $\mathbb{C}^n$ ($n>1$) ? I would be pleased if you tell me.
user135645's user avatar
9 votes
1 answer
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Automorphisms of $GL_n(\mathbb{Z})$

I want to consider the crossed module: $H \xrightarrow{t} Aut(H)$ for the case where $H = GL_n(\mathbb{Z}) = Aut(T^n)$ is the automorphism group of the $n$-torus. Any suggestions on how to understand ...
cheyne's user avatar
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3 votes
0 answers
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Reference of generalized isometries

I'm wondering if these objects have a name or are studied. No one around me knows, so I thought to ask here. Let $\Phi:\mathbb{R}^d\rightarrow \mathbb{R}^d$ be $C^2$-diffeomoprhism, and fix $p \geq ...
ABIM's user avatar
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Is there a name of semidirect product of a group with its automorphism group?

Consider the construction $G \rtimes \text{Aut}(G)$. Here $ G$ is a group, $\text{Aut}(G)$ is the automorphism group and the semidirect product is over the most obvious action. 1) Is there any name ...
Breakfastisready's user avatar
3 votes
1 answer
250 views

How large can a symmetric generating set of a finite group be?

Let $G$ be a finite group of order $n$ and let $\Delta$ be its generating set. I'll say that $\Delta$ generates $G$ symmetrically if for every permutation $\pi$ of $\Delta$ there exists $f:G\...
Pavel Madaj's user avatar
0 votes
0 answers
89 views

Functoriality of indiscernible sequences

Let $T$ be a first order theory of, say, some type of combinatorial geometries which contain indiscernible sequences of points. Let $(\Gamma,\mathcal{O})$ be a model of $T$, where $\Gamma$ is the ...
THC's user avatar
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18 votes
2 answers
557 views

Can the graph of a symmetric polytope have more symmetries than the polytope itself?

I consider convex polytopes $P\subseteq\Bbb R^d$ (convex hull of finitely many points) which are arc-transitive, i.e. where the automorphism group acts transitively on the 1-flags (incident vertex-...
M. Winter's user avatar
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4 votes
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226 views

Groups inducing edge-colorings on graphs. Is this concept known?

Are the following concepts known in graph/group theory, and if Yes, what are they called and where to read about them? Because I do not know better, I gave them placeholder names for now. 1. ...
M. Winter's user avatar
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9 votes
2 answers
827 views

Reference request: birational automorphism group is finite

I am interested in having a look at the proof of the following fact: If $X$ is a smooth variety of general type, then $\mathrm{Aut(X)}$ is finite. I know that this is proved in "On algebraic groups ...
Stefano's user avatar
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3 votes
1 answer
88 views

A partition of the set of order 2 outer automorphisms of $SU(N)$

Let $N$ be an even integer, $N>2$. Let $E$ be the set of all outer automorphisms $\phi$ of $G = SU(N)$ which are of order 2, i.e. $\phi \circ \phi = \mathrm{id}_G$. Choose a particular element $\...
Antoine's user avatar
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7 votes
2 answers
1k views

What are the automorphism groups of direct products of dihedral group D4

What is the automorphism group of direct sum of dihedral group of order $8$, $D_4$? For example, $\mathrm{Aut}(D_4)$ is isomorphic to $D_4$. How about $\mathrm{Aut}(D_4\times D_4)$, $\mathrm{Aut}(...
Sirui Lu's user avatar
4 votes
1 answer
601 views

Schreier conjecture -- without a simple proof? and sporadic simple groups

The Schreier conjecture asserts that $\mathrm{Out}(G)$ is always a solvable group when $G$ is a finite simple group. This result is known to be true as a corollary of the classification of finite ...
wonderich's user avatar
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10 votes
0 answers
431 views

On the Number of Parallel Automorphism Lines

Given a group $G$, one can define the transfinite line of iterative automorphisms of $G$ to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the ...
Morteza Azad's user avatar
10 votes
1 answer
1k views

The Tall Tale of Terminating Transfinite Towers

The transfinite tower of iterative automorphisms of a group $G$ is simply definied to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the direct ...
Morteza Azad's user avatar
1 vote
1 answer
201 views

Lie brackets of automorphisms

Let $F$ be the vector fields of a differential manifold $M$, let $[X,Y]$ be the Lie brackets of $F$, now let $a$ be an automorphism of $F$ for the structure of real vector space of $F$. I consider now ...
Antoine Balan's user avatar
13 votes
0 answers
332 views

An example of curves with the same Jacobian, but different Jacobian automorphism groups (wrt their respective canonical principal polarizations)?

I am trying to understand examples of differing curves with the same Jacobian, and the quirks of the Jacobian. Here is my question: What is an example where $Aut(Jac(X, a))$ and $Aut(Jac(X', a'))$ ...
Catherine Ray's user avatar
4 votes
1 answer
168 views

automorphisms of a measurable space can be approximated by continuous measure preserving maps?

Suppose first that $D=[0,1]$ is equipped with the usual Lebesgue measure, and that $\varphi$ is a measure-preserving transformation $\varphi:D\to D$ that is bijective and whose inverse is also measure ...
Exodd's user avatar
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7 votes
1 answer
463 views

$\operatorname{Out}(F_n)$ is not linear for $n > 3$

The paper The Tits alternative for $\operatorname{Out}(F_n)$ I by Bestvina, Feighn and Handel and the paper Automorphisms of free groups and Outer space by Vogtmann both state that $\operatorname{Aut}(...
Student's user avatar
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5 votes
0 answers
119 views

Sign preserving Galois automorphisms

I have an algebraic number $\alpha \in \mathbb{Q}(\zeta)$, where $\zeta^n = 1$ is a root of unity (not primitive) given as a linear combination of powers of $\zeta$, i.e, $\alpha = \sum_{i=1}^k a_i \...
mathstudent42's user avatar
10 votes
1 answer
509 views

The Tits alternative for $\operatorname{Out}(F_n)$

Not sure if this is the right place to ask this, but the paper I am reading seems to be too specialised for mathstack (if you do not agree, pleas let me know and I will take down this question) I am ...
Student's user avatar
  • 275
4 votes
2 answers
473 views

How do eigenvalues of combinatorial Laplacian relates to automorphisms in graphs?

Is there a relation between eigenvalues of the graph Laplacian and the automorphism group of a simple graph? How are the multiplicities of Laplacian eigenvalues related to the order of the ...
linello's user avatar
  • 171
12 votes
2 answers
1k views

Graph automorphism group

Let $A_w$ denote such set of positive integer $n$ that: for any two permutations $\pi_0,\pi_1\in S_n$, if $\pi_1$ is not a power of $\pi_0$, then there exists a (labeled non oriented) graph $G$ of ...
Jiayi Liu's user avatar
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9 votes
2 answers
426 views

How to characterize "matching-transitive" regular graphs?

I am interested in regular graphs $G$ such that for each pair of 1-factors (=perfect matchings) $F$ and $F'$ there is an automorphism of $G$ that takes $F$ to $F'$. Let's call this property matching ...
Wolfgang's user avatar
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7 votes
3 answers
427 views

When do automorphisms on open subsets extend

Let $X$ be a normal affine variety of dimension at least two over $\mathbb{C}$ and let $U\subset X$ be a dense open. Assume that $\mathrm{codim}(X\setminus U) \geq 2$. I think Hartog's lemma implies ...
Xiboto's user avatar
  • 283
18 votes
3 answers
766 views

If a variety $X$ has finite automorphism group, is the same true for its $n$-fold self-products?

Let $X$ be an algebraic variety over $\mathbb{C}$. Let $n\geq 1$ be an integer and let $X^n$ be the $n$-fold self product of $X$. Q. Is there an integer $n\geq 1$ and an algebraic variety $X$ ...
Xiboto's user avatar
  • 283
14 votes
4 answers
669 views

Non-split Aut(G) $\to$ Out(G)?

I'm looking for examples of outer automorphisms of a finite group $G$ which do not lift to automorphisms (i.e. non-split quotient map $\mathrm{Aut}(G)\to \mathrm{Out}(G)$, where $\mathrm{Out}(G) = \...
Kevin Walker's user avatar
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4 votes
0 answers
198 views

Is finiteness of automorphism groups a birational question?

Let $X$ be a proper variety over $\mathbb{C}$ and let $Y\to X$ be a proper birational surjective morphism. Is $Aut(X)$ finite if and only if $Aut(Y)$ is finite? The answer is no. Indeed, $\mathbb{P}^...
Elliot's user avatar
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2 votes
2 answers
189 views

question on simple groups [closed]

Let $G$ be a simple finite group. Then every non trivial endomorphism of $G$ is an automorphism. My question is: does the converse holds? More precisely, if $G$ is a finite group all of whose non ...
haily's user avatar
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4 votes
0 answers
141 views

Automorphisms of the modular product of graphs

Given graphs $X$ and $Y$, their modular product, which I will denote by $X \diamond Y$, has vertex set $V(X) \times V(Y)$ where two vertices $(x,y)$ and $(x',y')$ are adjacent if ($xx' \in E(X)$ and $...
David Roberson's user avatar
1 vote
1 answer
323 views

Automorphism group of fiber products of schemes

Let $A \mapsto S$ and $B \mapsto S$ be two schemes over the scheme $S$. Is there a connection between the automorphism group of the scheme $A \otimes_{S} B$ and the automorphism groups of $A$ and $B$ ?...
THC's user avatar
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14 votes
1 answer
495 views

Birational automorphisms of varieties of Picard number one

Let $X$ be a smooth projective variety of Picard number one, and let $f:X\dashrightarrow X$ be a birational automorphism which is not an automorphism. Must $f$ necessarily contract a divisor?
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