# 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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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:... 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 ...
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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 ... 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 ...
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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 ...
1 vote
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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)=\...
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 ...