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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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### 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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### 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 ...
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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