# Questions tagged [automorphisms]

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

69
questions

4
votes

1
answer

157
views

### Subfields of division rings of degree $2$ which are not invariant

Let $A$ be a noncommutative division ring, and let $B$ be a sub division ring (here, $B$ is allowed to be commutative) of degree $2$. Are there easy examples known for which $B$ is not globally fixed ...

1
vote

0
answers

38
views

### Twisting a graded algebra by an automorphism (Transitivity)

Definition: Let $A=A=\bigoplus_{j=0}^{\infty} A_j$ be a connected $\mathbb{N}$-graded $k$-graded algebra and let $\phi\in\text{Aut($A$)}$ be a graded automorphism of degree zero. A new graded algebra ...

3
votes

1
answer

261
views

### The $1$-dimensional Jacobian Conjecture over $\mathbb{Z}$-torsion free rings

I am looking for further proofs, preferably in the literature, of the following result:
Proposition: Let $R$ be a unital commutative $\mathbb{Z}$-torsion free ring. If $f(x) \in R[x]$ with $f'(x) \in ...

2
votes

0
answers

51
views

### Composition of correspondences pulled back to $\mathrm{CH}_0$

Let $X,Y,Z$ be varieties. Given two correspondences $\Gamma_1 \subset X \times Y$ and $\Gamma_2 \subset Y \times Z$ there is a composition,
$$ [\Gamma_1] \circ [\Gamma_2] = \pi_{13 *} (\pi_{12}^* [\...

0
votes

0
answers

85
views

### What is $(C, D, \delta, \gamma)$ and $(C, \delta; D, \gamma)$ Desarguesian?

A projective plane is $(C, \gamma)$-Desarguesian if for any 2 triangles $A_1 B_1 C_1, A_2 B_2 C_2$ in perspective from $C$ (which means $C \in A_1 A_2, B_1 B_2, C_1 C_2$) such that $A_1 B_1 \cap A_2 ...

4
votes

1
answer

266
views

### Automorphisms of vector spaces and the complex numbers without choice

In Zermelo-Fraenkel set theory without the Axiom of Choice (AC), it is consistent to say that there are models in which:
there are vector spaces without a basis;
the field of complex numbers $\mathbb{...

1
vote

0
answers

108
views

### Is the commutator of the holomorph of generalized quaternion group abelian?

Let $Q_{2^{n}} = \langle x, y \mathrel\vert x^{2^{n-1}}=y^4 = 1, x^{2^{n-2}}=y^2, y^{-1}xy = x^{-1} \rangle$ be the generalized quaternion group of order $2^{n}$.
Let $\operatorname{Hol}(Q_{2^{n+1}})$ ...

3
votes

0
answers

51
views

### Set of equivalence classes of a Lie algebra under the action of the automorphism group

I recently became interested in the following question: Given a Lie algebra $\mathfrak{g}$, define two elements $x,y\in\mathfrak{g}$ to be equivalent if there exists an automorphism $\phi\in\...

2
votes

1
answer

109
views

### About external automorphism on non-well founded model of Finite ZF?

Let $M$ a non-well founded model of Finite $\sf ZF$, which is $\sf ZF$ with axiom of infinity replaced by the axiom stating that all sets are finite. So there must be a set $\zeta$ that $M$ thinks it'...

2
votes

0
answers

165
views

### Automorphisms of a K3 surface

I was studying the following algebraic surface in $\mathbb{P}^5$ defined by the following three quadrics:
\begin{cases}
x^2 + xy + y^2=w^2\\
x^2 + 3xz + z^2=t^2\\
y^2 + 5yz + z^2=s^2.
\...

3
votes

0
answers

64
views

### Field automorphisms of projective spaces without the axiom of choice

Suppose P is a projective space over the field $k$. If P has finite dimension $n$, we can fix a base. Relative to this base, the full automorphism group of P can be described by the action on the ...

2
votes

1
answer

231
views

### $K3$ surfaces in $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$

I am considering $K3$ surfaces in $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$ with an automorphism that preserves an ample divisor class.
For an automorphism $\rho$ of a $K3$ surface, let ${\...

4
votes

0
answers

78
views

### Curves not invariant by non-trivial projective automorphisms

Let $g\ge 0$, $d\ge 1$ be integers. We consider the Moduli space $H_{g,d}$ parametrising smooth irreducible closed curves $C\subset \mathbb{P}^3$ of degree $d$ and genus $g$. Let us denote by $U_{g,d}$...

16
votes

2
answers

781
views

### Groups which maintain all their subgroups’ automorphisms as inner automorphisms

Are there any groups, finite or infinite, other than the first three symmetric groups which maintain all their subgroups’ automorphisms as inner automorphisms (every automorphism of every subgroup ...

1
vote

0
answers

80
views

### Are these maps, associated to finite simple graphs, interesting?

Given a finite simple graph on $n$ vertices, say $G = (V,\, E)$, where
$$ V = \{ v_1, \ldots , \, v_n \} $$
and
$$ E \subseteq \{ (v_a, \, v_b) \, | \, 1 \leq a < b \leq n \},$$
does there exist a ...

-2
votes

1
answer

149
views

### Are isomorphic quotients of abelian groups induced by automorphisms? [closed]

If I have an (abelian) group $G$ and an automorphism $\sigma: G \to G$ then for any subgroup $H$ of $G$ there is an induced isomorphism $G/H \cong G/\sigma(H)$ given by the map $gH \mapsto \sigma(g)\...

0
votes

1
answer

164
views

### Moving general fibers of a fibration

Let $X$ be an irreducible projective variety over $\mathbb{C}$ admitting a morphism $\pi:X\rightarrow \mathbb{P}^1$ with connected fibers. We may assume that the general fiber of $\pi$ is smooth.
My ...

1
vote

1
answer

135
views

### Would automorphisms cause nested subset-hood?

Working in $$\sf ZF + GC + j:V \xrightarrow {auto} V + \exists \alpha: V_{j(\alpha)} \subsetneq V_\alpha \land \alpha \text{ is limit }$$
Of course $j$ is external in the sense that it is not used in ...

1
vote

0
answers

65
views

### Is existence of external rank shifting automorphism proves moving of infinitely ranked stratified power stages of this theory?

In this posting, I've define stratified power sets $\mathcal P^\equiv$ operator.
Now we define $V^\equiv_\alpha$ as the iterative stratified power sets of $V_\omega$ as:
$$V^\equiv_0 = V_\omega \\ V^\...

1
vote

1
answer

142
views

### Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?

Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?
The examples of rings not isomorphic to their opposite that I know of are not ...

1
vote

1
answer

90
views

### When is a infinite transcendence-degree rigid fields fixed by a finite extension?

A field is rigid if it has no nontrivial automorphisms. Let $F$ be a rigid field which has infinite transcendence degree over $\mathbb{Q}$, and let $E$ be a finite extension of $F$. Then my question ...

5
votes

1
answer

264
views

### Isomorphisms of complete intersections

Let $X, Y\in \mathbb{P}^n$ be two singular Fano complete intersections of the same multidegree $(d_1,…,d_r)$.
If we assume there is an isomorphism $f\colon X\rightarrow Y$ are there any assumptions so ...

4
votes

1
answer

171
views

### Certain endomorphisms of $\mathbb{C}(x,y)$

Let $f: (x,y) \mapsto (p,q)$ be a $\mathbb{C}$-algebra endomorphism of $\mathbb{C}(x,y)$
satisfying the following two conditions:
(i) $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$.
(...

1
vote

0
answers

97
views

### Action on cohomology by automorphisms of ihs manifolds

For all known deformation types of irreducible holomorphic symplectic manifolds (which I call K3, K3$^n$, Kum$^n$, OG$^3$, OG$^5$, the exponent being half the complex dimension), it is known that the ...

3
votes

0
answers

127
views

### Isomorphisms of weighted complete intersections

Let $X\subset\mathbb{P}(a_0,\dots,a_n)$ and $Y\subset\mathbb{P}(b_0,\dots,b_n)$ be two weighted complete intersections with mild (say terminal) singularities.
Assume that there is an isomorphism $f:...

9
votes

1
answer

377
views

### Automorphisms and epimorphisms of finite groups

All groups in this question are finite, and epimorphism means surjective group homomorphism.
Suppose I have two epimorphisms $f,g\colon G\to H$. This implies that $\ker(f)$ and $\ker(g)$ have the ...

3
votes

1
answer

381
views

### Automorphisms of singular hypersurfaces

Let $X\subset\mathbb{P}^{n+1}$ be an irreducible and reduced hypersurface of degree $d$.
A theorem by Matsumura and Monski asserts that if $n\geq 2$, $d\geq 3$, $(n,d)\neq (2,4)$ and $X$ is smooth ...

2
votes

0
answers

116
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

0
answers

302
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

1
answer

492
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

1
answer

184
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

0
answers

59
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

0
answers

134
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

1
answer

221
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

1
answer

192
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

3
answers

567
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

0
answers

181
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 ...

1
vote

2
answers

525
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,...

5
votes

1
answer

431
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

1
answer

193
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

1
answer

320
views

### Automorphism group of tensor product of two graphs

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 ...

8
votes

1
answer

407
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

0
answers

123
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

0
answers

151
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

1
answer

410
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 ...

0
votes

2
answers

287
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 ...

3
votes

1
answer

433
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 ...

0
votes

2
answers

314
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

1
answer

450
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 ...

5
votes

0
answers

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{...