# 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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### $G/F(G)$ is isomorphic to $X_1\times\cdots\times X_t$

I asked this question on math.stackexchange many hours ago, but haven’t got an answer. It was mentioned in a comment that the answer to my question is trivial, but I couldn’t see why. $G$ is a finite ...
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### What can we get from an automorphism of order $2$

An automorphism of order $2$ is an automorphism fixes some elements and inverts the others. It’s well known that not all groups have automorphisms of order $2$, $C_2$, for example. But if a group ...
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### Outer automorphism group of $F(G)$

By a nice helpful comment in my last question, I see that if $\Phi(G)=1$ then ${\rm Out}(F(G))$($\cong G/F(G))$) is isomorphic to a direct product of ${\rm GL}(n_i,p_i)$. Actually, I’m digesting an ...
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### Fitting subgroup of a solvable group is a direct product of some elementary abelian $p$-groups

I saw a remark saying If $G$ is solvable, then ${\rm Out}(F(G))$ is isomorphic to a direct product of ${\rm GL}(n_i,p_i)$ (except for certain value of $n_i$ and $p_i$). I know the key is to prove ...
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### Why $G/F(G)$ is isomorphic to a subgroup of ${\rm Out}(F(G))$?

I know two facts and I’ve managed to figure out how to prove one, but the other one is still a little confusing. Let $G$ be a finite solvable group and $F(G)$ is the Fitting subgroup of $G$. (...
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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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### Small automorphism groups of groups

I do not know much about group theory, so sorry in case this question is not for MO. For a finite group $G$, denote by $f(G)$ the number of elements of the automorphism group of $G$. Question: For ...
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### English translation of Fouxe-Rabinovitch paper

Is there somewhere an english translation of Fouxe-Rabinovitch's papers "D. I. Fouxe-Rabinovitch, Uber die Automorphismengruppen ¨ der freien Produkte. II, Rec. Math. [Mat. Sbornik] N.S., 1941, ...
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### Can the automorphism group vary too much in families of complex projective varieties?

In a family of smooth projective curves over a reduced complex scheme of finite type the list of isomorphism classes of automorphism groups of the fibers is finite. This follows from the Hurwitz's ...
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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 ...
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