Questions tagged [automorphisms]
An automorphism is an isomorphism from an object to itself, and which also preserves the objects structure .
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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 ...
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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 ...
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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 ...
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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}^* [\...
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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 ...
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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{...
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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}})$ ...
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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\...
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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'...
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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.
\...
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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 ...
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$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 ${\...
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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}$...
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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 ...
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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 \},$$
does there exist a ...
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Are isomorphic quotients of abelian groups induced by automorphisms? [closed]
If I have an (abelian) group $G$ and an automorphism $\sigma: G \to G$ then for any subgroup $H$ of $G$ there is an induced isomorphism $G/H \cong G/\sigma(H)$ given by the map $gH \mapsto \sigma(g)\...
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Moving general fibers of a fibration
Let $X$ be an irreducible projective variety over $\mathbb{C}$ admitting a morphism $\pi:X\rightarrow \mathbb{P}^1$ with connected fibers. We may assume that the general fiber of $\pi$ is smooth.
My ...
user114666
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Would automorphisms cause nested subset-hood?
Working in $$\sf ZF + GC + j:V \xrightarrow {auto} V + \exists \alpha: V_{j(\alpha)} \subsetneq V_\alpha \land \alpha \text{ is limit }$$
Of course $j$ is external in the sense that it is not used in ...
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Is existence of external rank shifting automorphism proves moving of infinitely ranked stratified power stages of this theory?
In this posting, I've define stratified power sets $\mathcal P^\equiv$ operator.
Now we define $V^\equiv_\alpha$ as the iterative stratified power sets of $V_\omega$ as:
$$V^\equiv_0 = V_\omega \\ V^\...
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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 ...
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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 ...
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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 ...
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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) $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$.
(...
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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 ...
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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:...
user61586
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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 ...
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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 ...
user125056
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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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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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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 $F$ ...
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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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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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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 ...
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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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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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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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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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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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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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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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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 ...
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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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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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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 ...
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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 ...
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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 ...
user58604
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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 ...
user56259
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2
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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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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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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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