Questions tagged [automorphisms]
An automorphism is an isomorphism from an object to itself, and which also preserves the objects structure .
1
vote
0answers
61 views
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 ...
5
votes
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 ...
7
votes
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
votes
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 ...
1
vote
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)...
3
votes
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
votes
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 $...
3
votes
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 ...
7
votes
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)=\...
2
votes
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 ...
2
votes
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,...
4
votes
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$...
1
vote
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$ ...
1
vote
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 ...
6
votes
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 ...
0
votes
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 ...
3
votes
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
vote
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
votes
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
votes
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 ...
1
vote
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 ...
12
votes
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 ...
4
votes
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{...
10
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 $X$.
...
11
votes
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 $G$ ...
2
votes
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
votes
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{...
15
votes
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 $\alpha \in Aut(A)$ ...
4
votes
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.
8
votes
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
votes
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
votes
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 ...
4
votes
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
votes
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 ...
