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      Questions tagged [automorphism-groups]

      Questions about the group of automorphisms of any mathematical object $X$ endowed with a given structure, i.e the group of all bijective maps from $X$ to itself preserving this structure, and hence helping study it further and understand it better.

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      What are the compact Aut(A) of an algebra A(G), G finite, that contains the identity?

      If we have an algebra A over a finite group G, then if G is non-abelian we can have a non-trivial set of compact automorphisms of A that map the elements of G onto a set isomorphic to G. It may be ...
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      50 views

      Automorphism group of the cycle graph with $k$ diagonals

      Let $C_n$ be the cycle graph on $n$ vertices. Then $Aut(C_n) \cong D_{2n}$ the dihedral group of order $n$. Now, let $C_{n,i}$ be the cycle graph with $i$ many diagonals connecting opposite vertices ...
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      62 views

      Are all verbal automorphisms inner power automorphisms?

      Let $G$ be a group. $\DeclareMathOperator{\Wa}{Wa}\DeclareMathOperator{\Tame}{Tame}\DeclareMathOperator{\Aut}{Aut}$ Let's call $\phi \in \Aut(G)$ verbal automorphism iff $\exists n \in \mathbb{N}, \{...
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      1answer
      226 views

      Infinite order automorphisms of planar polynomials

      Let $R_n$ be the integral polynomial ring $\mathbb{Z}[x_1,x_2,...,x_n]$, let $A_n$ be the group of ring automorphisms $\mathrm{Aut}(R_n)$, and for $f\in R_n$ let $\mathrm{Aut}(f)=\{\alpha\in A_n\ |\ \...
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      1answer
      173 views

      Abelian torsion-free group with $\mathbb{Z}_2\times\mathbb{Z}$ as automorphism group

      Let $A$ be an abelian torsion-free group. If $A$ is isomorphic with the group of rational numbers whose denominators are powers of, say, $2$, then its automorphism group is isomorphic with $\mathbb{Z}...
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      1answer
      270 views

      Automorphism group of the special unitary group $SU(N)$

      Let us consider the automorphism group of the special unitary group $G=SU(N)$. We know there is an exact sequence: $$ 0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0. $$ For $G=SU(2)...
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      176 views

      Automorphism group of poset of number fields

      Consider the poset of number fields, partial order being defined by inclusion of fields. What is the group of order-preserving automorphisms of this poset? What if we take only Galois extensions of $\...
      4
      votes
      1answer
      132 views

      Metrics with prescribed Levi-Civita connection

      My question involves the symmetries of a (pseudo)-Riemannian metric preserving the Levi-Civita connection (LCC), its unique torsion-free metric connection. For a basic example, one notes that the ...
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      41 views

      Does similar automorphism group imply similar extension complexity?

      Definition: Extension complexity of a polytope $P$ in $\mathbb R^n$ is the minimal number of facets a polytope in $\mathbb R^{n'}$ with $n'\geq n$ needs such that its projection is $P$. Do there ...
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      1answer
      71 views

      Sabidussi theorem for morphisms between graphs

      Sabidussi proved that if a finite graph $X$ is isomorphic to a Cartesian product of connected graphs $X_1,\ldots,X_m$ which are pairwise relatively prime with respect to Cartesian multiplication, then ...
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      1answer
      181 views

      What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in C^n ( n>1)? [closed]

      What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in $\mathbb{C}^n$ ($n>1$) ? I would be pleased if you tell me.
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      1answer
      269 views

      Automorphisms of $GL_n(\mathbb{Z})$

      I want to consider the crossed module: $H \xrightarrow{t} Aut(H)$ for the case where $H = GL_n(\mathbb{Z}) = Aut(T^n)$ is the automorphism group of the $n$-torus. Any suggestions on how to understand ...
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      0answers
      76 views

      Reference of generalized isometries

      I'm wondering if these objects have a name or are studied. No one around me knows, so I thought to ask here. Let $\Phi:\mathbb{R}^d\rightarrow \mathbb{R}^d$ be $C^2$-diffeomoprhism, and fix $p \geq ...
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      1answer
      430 views

      Is there a name of semidirect product of a group with its automorphism group?

      Consider the construction $G \rtimes \text{Aut}(G)$. Here $ G$ is a group, $\text{Aut}(G)$ is the automorphism group and the semidirect product is over the most obvious action. 1) Is there any name ...
      3
      votes
      1answer
      182 views

      How large can a symmetric generating set of a finite group be?

      Let $G$ be a finite group of order $n$ and let $\Delta$ be its generating set. I'll say that $\Delta$ generates $G$ symmetrically if for every permutation $\pi$ of $\Delta$ there exists $f:G\...

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