# Questions tagged [automorphism-group]

For finding, constructing and proving results about automorphisms and applying them into different contexts. An automorphism is an isomorphism from an object to itself, and collectively they form a group under composition of mappings. Sometimes the only automorphism is the trivial one (the identity map), but often structures will have interesting/non-trivial automorphisms.

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### What's an automorphism of a function?

I have as a theorem that the automorphism group of a function is homomorphic to $\Bbb Z/2\Bbb Z$. While it is fairly clear to me what this means im my specific case, I'm unclear what it means in ...
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### Meaning of the terms "operation" and "invariant" in the old group theory paper

I am reading an old paper C.Hopkins, "Non-abelian groups whose groups of isomorphisms are abelian", 1928.(Link: 1) I don't know what the term "operation" and "invariant" ...
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### Extending an isomorphism between subfields of an algebraically closed field

Let $k$ and $k'$ be subfields of an algebraically closed field $\ell$, and suppose $\sigma: k \mapsto k'$ is an isomorphism of fields. When can $\sigma$ be extended to an automorphism of $\ell$ (upon ...
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### Proof of $\operatorname{Aut}_G(G/H)\cong N_G(H)/H$

Here $G$ is a group with subgroup $H$, and we let $G$ act on $G/H$ by left multiplication. Correspondingly, $G/H$ is a left $G$-set and the set $\operatorname{Aut}_G(G/H)$ denotes the set of all $G$-...
1 vote
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### What's the automorphism group of $ax-b\cdot2^{\nu_2(x)}: a,b\in\Bbb Z_2^\times$ on $\Bbb Q_2$?

Let $f_{a,b}(x)=ax-b\cdot2^{\nu_2(x)}$ act on the 2-adic numbers. Then I have as a fairly resolute theorem that for any choice of two 2-adic units $a,b$, every $f_{a,b}$ is topologically conjugate to ...
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### Automorphisms with order 2

Let T be an automorphism of finite group G & order of T is 2. Taking any element in G say x then $$xT = xT^{-1} = x^{-1}T$$ which implies $x=x^{-1}$ by taking $xTT^{-1} = x^{-1}TT^{-1}$. But since ... 47 views

### the non-abelian subgroups of the Lamplighter group

The lamplighter group can be defined by the semidirect product: $$L_2=(\mathbb{Z} _2) \wr \mathbb{Z} \cong \bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_{2} \rtimes_\phi\mathbb{Z},$$ where $\phi(1)$ &...
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1 vote
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### Relation of the mapping class group and the symplectic group.

Generally, I am currently investigating automorphisms of Riemann surfaces and their non-trivial action on the first homology. I wonder whether one can make explicit statements on the relationship ...
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### Automorphism group of an R-module

Let R be a commutative ring. It is well known that the automorphism group of the module $R^n$ is isomorphic to $GL_n(R)$. Is their a way to measure how much this fails for an arbitrary module M? ...
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### Galois group of an arbitrary field extension

It is well known that if $E/F$ is a finite field extension, then $|Gal(E/F)|\leq [E:F]$. One can also prove this for extension with $|Gal(E/F)|$ is countable, by Theorem 13, Page 36 of [E. Artin, ...
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### What is this notation for a sum?

This sum is written in my Galois Theory notes. Surely if $a_{{i_1}{i_1}...{i_t}}$ then the $i_1$'s would just infinitely repeat, along with the $i_2$'s, etc. This notation just doesn't compute with ...
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### Show that every automorphism of $F_{20} = \langle a, b \mid a^5 = b^4 = 1, bab^{-1} = a^3\rangle$ is an inner automorphism

Show that every automorphism of $F_{20} = \langle a, b \mid a^5 = b^4 = 1, bab^{-1} = a^3\rangle$ is an inner automorphism. I found $Z(F_{20})={1}$ and it follows $F_{20} \cong \text{Inn}(F_{20})$. ...
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### Properties of the obvious action of $Aut(G)$ on $G$

Question: Let $G$ be a finite group. Let $Aut(G)$ be the group of automorphisims of $G$. Consider the group action $\phi:Aut(G)\times G\to G$ where $\phi(\sigma,g)=\sigma(g)$. Assume $G$ has exactly ...
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### Show that in a group $G$ of order $165=3\cdot 5\cdot 11$, the center $Z(G)$ contains a subgroup of order 11.

I have solved most of the problem, but I still can't figure out the last part. If $P_{11}\in Syl_{11}(G)$, then $n_{11}\equiv 1 \pmod{11}$ and $n_{11} \mid 3\cdot 5$, i.e. $n_{11}=1$, so that $P_{11}$ ...
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### Abelian subgroups of the group of automorphisms of a finite group

This is a follow-up question from my post here, which has been moved according to a comment. For context, here is the setup. Let $G$ be a nontrivial finite group. In his book "Finite Group ...
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### Show that $G$ is a complete or empty graph on n vertices if and only if every transposition of $\{1, 2, . . . , n\}$ belongs to $Aut(G)$.

Here $Aut(G)$ is the set of all Automorphisms of the graph $G$. One direction of the proof is easy as for all empty or complete graph, every permutation of the vertex set will belong to the set of ...