# Questions tagged [automorphisms]

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

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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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### 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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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 $gH \mapsto \sigma(g)\... • 13 0 votes 1 answer 176 views ### Moving general fibers of a fibration Let$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 136 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 ... • 10.5k 1 vote 0 answers 66 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^\... • 10.5k 1 vote 1 answer 146 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 99 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 ... • 4,547 5 votes 1 answer 298 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 176 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\}. (... • 2,767 1 vote 0 answers 104 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 135 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 411 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 ... • 55.7k 3 votes 1 answer 450 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 124 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 ... • 13.1k 5 votes 0 answers 306 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 ... • 599 7 votes 1 answer 503 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. $$We say that a permutations \psi of F ... 3 votes 1 answer 206 views ### Extension of isomorphism of fields I know this is not a "research Mathematical question", but this is a question that I would ask to my Math colleagues, but no one could give me a readily "yes-no" answer. Unfortunately, this is not a ... 1 vote 0 answers 60 views ### Invertibility under base change for the Weyl algebra instead of for the polynomial algebra From Lemma 1.1.8, we obtain the following: Assume that R \subseteq S are commutative rings and u: R[x,y] \to R[x,y] is an R-algebra endomorphism that has an invertible Jacobian, namely, Jac(u(x)... • 2,767 3 votes 0 answers 138 views ### What is the name of this substructure/embedding? I am interested in the following property, be it on an abstract or concrete category: A is a substructure of B such that every automorphism of A extends uniquely to an automorphism of B. Or ... 5 votes 1 answer 244 views ### Permutations of points in the projective plane Let p_1,...,p_7\in\mathbb{P}^{2} be seven general points in the projective plane \mathbb{P}^{2} over the complex numbers. Let f be an automorphism of \mathbb{P}^{2} inducing a permutation of ... • 51 3 votes 1 answer 205 views ### Extending an automorphism to an inner one Let D be a division ring. I have in mind the following result. Theorem. For every automorphism f of D, there is a division ring E extending D such that f extends to an inner automorphism ... • 1,555 7 votes 3 answers 614 views ### growth of a free group automorphism is same for finite index subgroups? Let X=\{x_1,\dots,x_N\} and F=F(X) be a free group generated by X. Let \phi\colon F\to F be an automorphism of F. Define a growth function of \phi as:$$ \operatorname{gr}_{\phi,X}(n)=\... • 1,050 2 votes 0 answers 213 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 ... • 4,533 1 vote 2 answers 583 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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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 $f$...