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Questions tagged [automorphisms]

An automorphism is an isomorphism from an object to itself, and which also preserves the objects structure .

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Are there half-transitive convex polytopes?

I only consider convex polytopes, i.e. convex hulls of finitely many points. The (edge-)graph of a polytope $P\subseteq\Bbb R^d$ is the graph consisting of the polytope's vertices, two are adjacent if ...
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0answers
275 views

Central extension of Tarski monsters

Suppose $G$ is a perfect group ($G=G'$) with the following properties. $G/Z(G)$ is a Tarski $p$-group or another simple finitely generated infinite group in which all proper subgroups are abelian, and ...
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1answer
392 views

Linear permutations commuting with $x\rightarrow x^{-1}$

Let $F = \operatorname{GF}(2^n)$ be a finite field. Define a permutation $\phi:F \rightarrow F$ by the formula $$ \phi(x) = x^{-1}, \ x\neq 0; \ \phi(0) =0. $$ We say that a permutations $\psi$ of $F$ ...
3
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1answer
157 views

Extension of isomorphism of fields

I know this is not a "research Mathematical question", but this is a question that I would ask to my Math colleagues, but no one could give me a readily "yes-no" answer. Unfortunately, this is not a ...
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0answers
40 views

Invertibility under base change for the Weyl algebra instead of for the polynomial algebra

From Lemma 1.1.8, we obtain the following: Assume that $R \subseteq S$ are commutative rings and $u: R[x,y] \to R[x,y]$ is an $R$-algebra endomorphism that has an invertible Jacobian, namely, $Jac(u(x)...
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0answers
111 views

What is the name of this substructure/embedding?

I am interested in the following property, be it on an abstract or concrete category: $A$ is a substructure of $B$ such that every automorphism of $A$ extends uniquely to an automorphism of $B$. Or ...
5
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1answer
151 views

Permutations of points in the projective plane

Let $p_1,...,p_7\in\mathbb{P}^{2}$ be seven general points in the projective plane $\mathbb{P}^{2}$ over the complex numbers. Let $f$ be an automorphism of $\mathbb{P}^{2}$ inducing a permutation of $...
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1answer
122 views

Extending an automorphism to an inner one

Let $D$ be a division ring. I have in mind the following result. Theorem. For every automorphism $f$ of $D$, there is a division ring $E$ extending $D$ such that $f$ extends to an inner automorphism ...
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3answers
300 views

growth of a free group automorphism is same for finite index subgroups?

Let $X=\{x_1,\dots,x_N\}$ and $F=F(X)$ be a free group generated by $X$. Let $\phi\colon F\to F$ be an automorphism of $F$. Define a growth function of $\phi$ as: $$ \operatorname{gr}_{\phi,X}(n)=\...
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0answers
150 views

Constructible sets, I (Morphisms)

I have a number of questions on constructible sets. The first one is on morphisms: suppose $X$ and $Y$ are constructible sets, respectively in projective spaces $\mathbf{P}_1$ and $\mathbf{P}_2$ over ...
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2answers
224 views

Cross-ratio and projective transformations

Let $P=\{p_1,\ldots,p_6\}\subset\mathbb{P}^1$ be a set of six general points of the projective line. In particular there are no two different subsets $\{p_{i_1},\ldots,p_{i_4}\}$ and $\{p_{j_1},\ldots,...
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1answer
217 views

Extending an automorphism from a sub-algebra to the algebra

Let $A \subseteq B$ be two (associative with $1$) $k$-algebras, where $k$ is a field of characteristic zero, and let $f$ be a $k$-automorphism of $A$. I am interested to know 'when' one can extend $f$...
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1answer
161 views

What happens when you internalize outer automorphisms?

Given a finitely presented group $G = (Gen|Rel)$, we have a set of inner automorphisms $\{ \phi_a(x) = axa^{-1} | a \in G\}$. Defining the set of outer automorphisms to be those automorphisms of $G$ ...
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0answers
97 views

automorphism group of tensor product of two graph

Is there any relation between the automorphism group of the tensor product of two graphs $G = G_1 \times G_2$ and the automorphism groups of $G_1$ and $G_2$? I am aware of the nice results for the ...
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1answer
195 views

Conditions for a finite group to be isomorphic to its automorphism group

So in the interest of gaining a better understanding of a conjecture (due to Scott, 1960) on the automorphism series (first part of the automorphism tower, no direct limits) of a finite group that ...
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0answers
103 views

Simple automorphisms of finite relations

Finding automorphisms is a hard problem in general, but I am studying some simple subgroups of automorphisms, which are easy to find. I have some r-ary relation R on a finite set U (if it was a ...
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0answers
121 views

Making extensions $L/K$ aware of the Galois group coming from $K/k$

Although inspired by my question on math.SE https://math.stackexchange.com/q/1902190/214353 this is not a crosspost. What happened is that after an answer and some comments I realized more clearly ...
1
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1answer
326 views

Automorphisms of rings fixing all prime ideals

Let $f,g:A \to B$ be two ring homomorphisms of noetherian rings satisfying that for any prime ideal $\mathfrak{q} \subset B$, $f^{-1}(\mathfrak{q})=g^{-1}(\mathfrak{q})=:\mathfrak{p}$ and the induced ...
2
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2answers
251 views

Curves in homogeneous varieties

Let $C$ be a curve in a projective homogeneous variety $X$. Fixed a general point $x$ in $X$, does there exist a curve $V$ in $X$ passing through $x$ and such that $C$ and $V$ have the same homology ...
5
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1answer
242 views

Automorphisms of Cartesian products

Let us consider the Cartesian product $X^r$, where $X$ is a smooth projective variety. There is a subgroup $Aut_{\Delta}(X^r)\subset Aut(X^r)$ of automorphisms of $X^r$ mapping a $k$-dimensional ...
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2answers
247 views

Automorphisms of locally trivial fibrations

Let $f:X\rightarrow Y$ be a locally trivial fibration with a variety $F$ as the fiber. Here $X, Y, F$ are smooth, projective varieties. Does any automorphism of $F$ induce an automorphism of $X$? In ...
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1answer
417 views

Obstructed automorphisms of schemes

Let $X$ be a smooth projective scheme over a field $\mathbf{k}$ of characteristic zero such that $\mathrm{H}^0(X, \mathrm{T}X)$ vanishes, and let $f$ be an automorphism of $X$. I would like to have an ...
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0answers
1k views

Examples of a topological semidirect product

Let $G$ be a compact topological group, and $\operatorname{Aut}(G)$ the group of autohomeomorphisms of $G$. I have proved some (topological) results about the holomorph $G\leftthreetimes \operatorname{...
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1answer
509 views

Automorphisms of del Pezzo surfaces

Let $S$ be a del Pezzo surface of degree six over $\mathbb{C}$. Then $S$ is the blow-up of $\mathbb{P}^2$ in three general points $p_1,p_2,p_3$. Is it true that its automorphism group is $((\mathbb{C}...
4
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1answer
368 views

Automorphisms of surfaces

Let $X$ be a projective surface with a morphism $f:X\rightarrow\mathbb{P}^1$. Assume that $f^{-1}(t)\cong\mathbb{P}^1$ for any $t\neq 0$ but $f^{-1}(0)$ is the union of two $\mathbb{P}^1$'s ...
0
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1answer
133 views

Graph automorphisms that preserve independent sets [closed]

Let $G=(V,E)$ be a graph and $\mathrm{Ind}(G)$ be the collection of its independent sets. We call a graph automorphism $f:V \to V$ of $G$ good if it is non-trivial and $f(\mathrm{Ind}(G))=\mathrm{...
5
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0answers
351 views

Very frustrated reading a proof of the faithfulness of Artin's representation of braid groups

I am reading BRAID GROUPS, FREE GROUPS, AND THE LOOP SPACE OF THE 2-SPHERE by F.R. Cohen and J. Wu and here is an extract of the paper: (The proof is not finished yet but I am very confused by now.) ...
2
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0answers
104 views

When polynomial GI implies polynomial (edge) colored GI?

(edge) colored graph isomorphism is GI which preserves the colors (of edges if it is edge colored). There are several reductions using transformations/gadgets of (edge) colored GI to GI. For edge ...
3
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1answer
885 views

How hard is a variant of graph automorphism problem?

I'm interested in a variant of graph automorphism problem (which is prime candidate for $NP$-Intermediate problem). Restricted GA Input: Given an undirected graph $G(E, V)$, and $\epsilon |V|/2$ ...
4
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1answer
183 views

Property of IA automorphisms of free groups

For $n \in\mathbb{N}$ define: $X_n=\{x_1,\ldots,x_n\}$, $F(X_n)$ the free group on $X_n$, $\varphi:F(X_n)\to F(X_{n-1})$ an epimorphism defined by $x_i\stackrel{\varphi}{\mapsto} x_i$ for $1\le i\le ...
4
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3answers
658 views

Automorphisms of a smooth quadric surface $Q\subset\mathbb{P}^{3}$

Let $Q\cong\mathbb{P^{1}_{1}}\times\mathbb{P^{1}_{2}}\subset\mathbb{P}^{3}$ be a smooth quadric surface. We have the following two actions on $Q$: $$S_2\times Q\rightarrow Q,\; (\sigma,(x,y))\mapsto\...
2
votes
1answer
285 views

why do automorphisms preserve ample divisors?

Let $X \hookrightarrow \mathbb{P}$ be a smooth hypersurface inside some projective space $\mathbb{P}$ and let $H$ be a smooth hyperplane section of $X$. Now let $\varphi$ be an automorphism of $X$. ...
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1answer
768 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 $G$ ...
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0answers
162 views

automorphisms of smooth complete intersection: action in cohomology

Let $X \subset \mathbb{P}^N$ be a smooth complete intersection, say over the complex numbers, and let $g$ be a finite order automorphism of $X$. I would like to prove that $g^\ast$ acts trivially on ...
2
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1answer
225 views

Can we flex the rigid models by enough power?

Definition (1): ‎An ‎‎$‎‎‎\mathcal{L}‎$ -‎ ‎structure ‎‎$‎‎‎\mathcal{M}‎$ ‎called ‎"‎‎rigid" ‎iff ‎‎there ‎is ‎no ‎non-trivial automorphism on ‎$‎\mathcal{M}‎$.‎ ‎‎ Definition (2): ‎An ‎‎$‎‎‎\mathcal{...
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2answers
653 views

automorphisms of C*-algebras and partial isometries

Let $A$ be a $C^*$-algebra, let $p$ and $q$ be Murray-von Neumann equivalent projections in $A$, i.e. there is a partial isometry $v$ such that $v^*v = p$ and $vv^* = q$. Suppose $\alpha \in Aut(A)$ ...
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1answer
348 views

Rigid Strongly Regular Graphs

I need a few examples of graphs that are strongly regular as well as rigid, i.e., have only the trivial automorphism. Any references to relevant literature would be appreciated. Thanks.
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1answer
273 views

Random variables invariant under almost automorphisms.

Let $\Omega$ be a standard atomless probability space, we can assume $\Omega=(0,1)$ with Lebesgue measure. A bijection $f:\Omega/A_1\to\Omega/A_2$ is almost automorphism, if $P(A_1)=P(A_2)=0$, $f(A)$ ...
12
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7answers
918 views

When does $Aut(X)=Bir(X)$ hold?

Let $X$ be a projective complex manifold. Under what condition do we have the equality $Aut(X)=Bir(X)$? Here $Aut(X)$ denotes the group of holomorphic automorphisms of $X$ and $Bir(X)$ the group of ...
2
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1answer
310 views

Isomorphism of connected, rigid, N-regular graphs with chromatic index N?

Background/Motivation I'm working on algorithms for canonical labeling of a certain class of graphs (motivated by biology). The "difficult" instances of this problem can be reduced to graphs of the ...
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0answers
129 views

which automorphisms of a subring extend to those of a ring

(Probably a silly question, but..) Consider the ring $R=k[[x_1,\dots,x_n]]/I$, (e.g. char(k)=0), and its subring, $R_1$, generated by some of $x_i$'s. In general, an automorphism of $R_1$ does not ...
2
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1answer
301 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 ...