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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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Small examples of non-transitive Automorphism groups of Steiner Systems

I'm currently doing research for a bachelor's seminar talk. I have found a result from E. Mendelsohn, "On the groups of automorphisms of Steiner triple and quadruple systems" stating that ...
dilemmma's user avatar
4 votes
1 answer
71 views

Why is Aut(Griess algebra) discrete?

Some algebras like the octonions have a continuous Lie group as their automorphism group e.g. $G_2$ According to Wikipedia the automorphism group of the Griess algebra is the Monster group. But that ...
zooby's user avatar
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4 votes
1 answer
320 views

Is there a general way to find the inverse of an automorphism of the free group? [closed]

If we describe an automorphism of the free group (on n generators) by where it sends the generators, is there some kind of algorithm to find the inverse automorphism? I am particularly interested in ...
Arlo Taylor's user avatar
12 votes
1 answer
188 views

Does there exist a group $G$ such that $\operatorname{Aut}(G)\cong D_5$, where $D_5$ denotes the dihedral group of order 10?

I came across this problem stated in the title having no clue what to do, and got stuck even in the finite case. Here's my attempt: I first proved that the group $G$, if it exists, cannot be finite ...
Cyankite's user avatar
  • 553
1 vote
0 answers
54 views

IsGroupOfAutomorphisms functionality

I'm looking for two things related to the GAP function IsGroupOfAutomorphisms: whether it does what I think it does (based on the brief GAP manual entry), and if so, how it works. The GAP manual ...
Michael Wynne's user avatar
3 votes
2 answers
164 views

How to find the order of $\text{Aut}(\text{Aut}(\mathbb{Z}_{1080}))$ [closed]

Is $\phi(\phi(1080)) = 96$ the solution? I understand that $\text{Aut}(\mathbb{Z}_n) \cong \mathbb{Z}^*_n$ from this link. Additionally, I guess that the order of $\text{Aut}(\mathbb{Z}_{1080})$ is ...
Chunibyo's user avatar
1 vote
0 answers
36 views

Proof of Thompsons $A \times B$-lemma

(Auxiliary lemma) Let $G$ be a $\pi$-group and $a$ a $\pi'$-element acting on $G$. If $X$ is a subnormal subgroup of $G$ with $[a, X] = 1 = [a, C_G(X)]$, then $[a, G] = 1$. Hey guys, I am having a ...
Stippinator's user avatar
2 votes
1 answer
36 views

Proving $N(\textrm{Cy}(G))=\textrm{Cy}(\textrm{Hol}(G))$

Let $G$ be a finite group, $\textrm{Hol}(G)$ a holomorph of $G$, and $\textrm{Cy}:\textrm{Hol}(G) \to S(G)$ a Cayley embedding. What I want to know is that the normalizer $N(\textrm{Cy}(G))$ in $S(G)$ ...
Akasa's user avatar
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1 vote
2 answers
138 views

If $G$ has automorphism $a \in Aut(G)$, that only fixes the identity and with $o(a)=2$, then $|G|$ is odd.

Let $G$ be finite group and $a \in Aut(G)$ with $o(a)=2$, if $a(g) \neq g$ for all $g \in G \setminus \{1\}$, then $|G|$ is odd. Hey Guys, I wanted to prove the Theorem above and was wondering if my ...
Stippinator's user avatar
2 votes
0 answers
58 views

When does a Lie algebra's outer automorphism group 'inherit' a representation?

In the following I am considering finite dimensional representations of semi-simple Lie algebras over fields of characteristic $0$. Examples should illuminate what I am getting at. Consider $\mathfrak{...
Craig's user avatar
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1 vote
2 answers
90 views

Question regarding the properties of an automorphism group of a Sylow P subgroup

The context for this question has to do with proving: Groups of order $pq$ with $p < q$ have a normal subgroup of order $q$ and are cyclic iff $q$ is not congruent to $1$ mod $p$. I will leave out ...
froitmi's user avatar
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1 vote
0 answers
40 views

Automorphism on the hyperreals

A field isomorphism $\phi:F\rightarrow G$ is a bijection such that (i) $\phi(x+y)=\phi(x)+\phi(y)$ and (ii) $\phi(xy)=\phi(x)\phi(y)$, where $F$ and $G$ are ordered fields. If $F=G$, then $\phi$ is ...
phst's user avatar
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1 vote
1 answer
79 views

Galois group of $Q(e^{2\pi i/14})/Q$

I want to know the Galois group of $Q(e^{2\pi i/14})/Q$. More precisely, I want to know how one can find the order of the Galois group and the automorphisms. First, I know that $x^{14}-1=(x^7 -1)(x^7 +...
Andrei's user avatar
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2 votes
0 answers
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Does a code being perfect have a specific effect on its automorphism group?

I know that the Perfect Binary Golay Code is very exceptional as it is the only perfect binary code that is not one of a few infinite families (Trivial, Simple Repitition, Hamming). The automorphism ...
nph's user avatar
  • 121
2 votes
2 answers
109 views

Application of theorem: Group with fixpointfree automorphism of order 2 is abelian.

Let $G$ be a group and $a \in Aut(G)$ with $o(a)=2.$ If $C_G(a)=1$, then $x^a=x^{-1}$ for all $x \in G$. In particular, $G$ is abelian. Hello, does anyone have an example where this theorem can be ...
Stippinator's user avatar
1 vote
1 answer
42 views

$Out(F_n)$ has a free abelian subgroup of rank $2n-3$

Proposition 9.5.4 The group $Out(F_n)$ has a free abelian subgroup of rank $2n-3$. This is a proposition of the book "Topological Methotods in Group Theory" by R. Geoghegan. In the proof he ...
Greg's user avatar
  • 422
2 votes
0 answers
27 views

Structure of automorphism groups of positive ternary quadratic forms

Let $f(x,y,z)=ax^2+by^2+cz^2+2ryz+2sxz+2txy$ be a positive definite integral ternary quadratic form (in Gauss's or Dickson's sense). Let $A(f)$ be its coefficient matrix in the usual sense. Then ...
Harun Kir's user avatar
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0 answers
41 views

automorphism group isomorphic to $\mathbb{Z}/2\mathbb{Z}$ [duplicate]

When is the automorphism group of a group $G$ isomorphic to $\mathbb{Z}/2\mathbb{Z}$ ? I noticed that the only groups whose automorphism group is trivial are trivial one and $\mathbb{Z}/2\mathbb{Z},$ ...
praton's user avatar
  • 1
5 votes
2 answers
272 views

Is every group the semidirect product of its center and inner automorphism group?

For every group $G$, we have $$G/Z(G)\simeq \operatorname{Inn}(G).$$ I wonder whether the quotient projection has a right inverse. I suspect it doesn’t have one in general. But I can’t find a counter ...
user760's user avatar
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1 vote
1 answer
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Exercise 7.2 in Algebraic Combinatorics by Stanley

This is exercise 2 in chapter 7 of Algebraic Combinatorics by Stanley. For part (a), I first found the entire automorphism group. By labeling the root 1, and then numbering off the remaining vertices ...
Jonathan McDonald's user avatar
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1 answer
72 views

Let $f \in \mathbb{A}ut(\mathbb{D)}$. Show that $\lim_{|z|\to 1}|f(z)|=1$.

I have to following problem to solve: Let $f \in \mathbb{A}ut(\mathbb{D)}$. Show that $\lim_{|z|\to 1}|f(z)|=1$. (Without using the precise form of the automorphism.) So far I've got the following ...
Felix U's user avatar
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0 answers
38 views

Group which acts properly on a tree is closed in the automorphisms group

Let $G$ be a locally compact group which acts properly on a locally finite (simplicial) tree $T$ (i.e., for each compact subset $K \subseteq T$ it holds that the set $G_K=\{g \in G| gK \cap K \neq \...
Bargabbiati's user avatar
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3 votes
2 answers
105 views

Common terminology for the set $\{\varphi(x) : \varphi \in \operatorname{Aut}(G)\}$ for an element $x$ of a group $G$

If $G$ is a group and $x \in G$, is there terminology referring to the orbit of $x$ under the action of the automorphism group? That is, the set $\{\varphi(x) : \varphi \in \operatorname{Aut}(G)\}$? ...
Robin's user avatar
  • 3,940
-1 votes
1 answer
62 views

Alternative definitions outer semidirect product [closed]

I am working with outer semidirect products and have come across two definitions that are slightly different. Let $N,H$ be two groups and $\theta: H \to \text{Aut}(N)$ a group morphism. Then in both ...
noparadise's user avatar
1 vote
1 answer
52 views

Realtionship between automorphism number of a graph and subgraph count

Suppose we have a simple, undirected, connected graph $G_N$ with $|V(G_N)| = N$ vertices and edge set $E(G_N)$. Graph has a adjacency matrix $\mathbf{A}_N = (a_{i,j})_{1\leq i,j\leq N}$, where $a_{i,j}...
Cantor_Set's user avatar
  • 1,082
3 votes
0 answers
24 views

Automorphism group of a direct product of directly indecomposable centerless groups.

I know that if $G$ and $H$ are two arbitrary groups $\mathrm{Aut}(G\times H)$ is not isomorphic in general to $\mathrm{Aut}(G)\times\mathrm{Aut}(H)$, in fact $\mathrm{Aut}(G)\times\mathrm{Aut}(H)$ can ...
Marcos's user avatar
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2 votes
0 answers
26 views

Simple parametrization of biholomorphism from a simple ellipse to the unit disk.

Let $D = \{ z \in \mathbb{C} \mbox{ with } |z|<1 \}$ be the open unit complex disk. Let $E_r= \{ z \in \mathbb{C} \mbox{ with } (\frac{\Re{z}}{r})^2 + (\Im{z})^2 < 1 \} $ be an open axis-...
ylvain's user avatar
  • 31
1 vote
0 answers
25 views

Simple parametrization of automorphisme (Biholomorphism) over a simple annulus

Let $D = \{ z \in \mathbb{C} \mbox{ with } |z|<1 \}$ be the open unit complex disk. Let $A_r = \{ z \in \mathbb{C} \mbox{ with } 1<|z|<r \}$ be an open annulus, centered at the origin, of ...
ylvain's user avatar
  • 31
1 vote
0 answers
64 views

Automorphism group of $x^3 - x - y^2$.

Let $k$ be some field of characteristic $\neq 2$. The polynomial $f(x,y) = x^3 - x - y^2$ is irreducible in $k[x,y] \cong k[x][y]$ as it cannot have a root in $k[x]$. By Gauss's lemma, $f$ is then ...
Lisa's user avatar
  • 138
2 votes
1 answer
73 views

Automorphism group of ${\rm PSL}_2(p)$ [closed]

This question is related to this answer by Prof. Holt. I can see why ${\rm PGL}_2(p)$ induces the full automorphism group of a Sylow $p$-subgroup $S$ of ${\rm PSL}_2(p)$. Let $a$ be any automorphism ...
Probability enthusiast's user avatar
1 vote
0 answers
50 views

Fundamental group of covering space as the kernel of homomorphism

Consider a surjective homomorphism $\theta: Z_2 * Z_3 \to S_3$ ($S_3$ is the symmetric group on 3 objects) given by mapping the generators to elements of order 2 and 3 in $S_3$ respectively. By ...
ricci_borel's user avatar
2 votes
0 answers
84 views

Automorphism group of $A_n$, $n \geq 7$ [duplicate]

I am trying to find the automorphism group of the alternating groups $A_n$. However, when it comes to $A_7$, I have found it difficult to prove that $\operatorname {Aut}(A_7) \cong S_7$. (I have ...
tys's user avatar
  • 163
1 vote
1 answer
49 views

Extending Special Automorphisms of Lie Manifolds

I call a smooth manifold $M$ special if it is diffeomorphic to a connected Lie group $G_M$ of automorphisms of $M$. Let $M$ be a special smooth manifold and $N$ a special smooth submanifold of $M$. $...
Jannis's user avatar
  • 197
5 votes
0 answers
93 views

Automorphic elements

Given two elements $x,y$ in a group, how can you quickly check whether there exists an automorphism $\phi$ such that $\phi(x) = y$? This seems well known but I can't find it online anywhere. ...
iurjfnee's user avatar
3 votes
2 answers
89 views

Show $\mathrm{Inn}(G)\,\operatorname{char}\,\mathrm{Aut}(G)$ for $G$ a non-abelian simple group

Let $G$ a non-abelian simple group and let $A=\mathrm{Inn}(G)$ and $B=\mathrm{Aut}(G)$. I would like to know the solution to $A\,\operatorname{char}\, B$. However, I know the following. Let $\phi \in \...
Akasa's user avatar
  • 71
1 vote
1 answer
56 views

Calculating fixed fields in $\mathbb{Q}$-automorphisms.

In an exercise, I am asked to determine the fixed subfield of an extension through given automorphism. I calculated them with the $\alpha$-method, I mean, writing a generic element of the extension ...
IAG's user avatar
  • 223
0 votes
0 answers
34 views

Calculating $[K(t):K\left(\dfrac{t^2}{t-1}\right)]$ with $t$ a trascendental element over $K$.

I was asked in an excercise of my fields and Galois theory course to calculate the degree of the extension $K(t)/K\left(\dfrac{t^2}{t-1}\right)$ with $t$ a trascendental over $K$. I started stuying ...
IAG's user avatar
  • 223
2 votes
0 answers
48 views

Looking for a group whose fixed field is given

Given a field $K$ and a transcendent element $t$, I'm asked to find both $$ \left[K(t):K\left(\frac{t^2}{t-1}\right)\right] \ \text{and} \ \text{Aut}\left(K(t)/K\left(\frac{t^2}{t-1}\right)\right) $$ ...
Valere's user avatar
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0 votes
0 answers
37 views

Finite group $G$ with $\alpha \in \text{Aut}(G)$ such that $x^{\alpha} \neq x = x^{{\alpha}^2}$ [duplicate]

The following question is from "The Theory of Finite Groups" by Hans Kurzweil. Let $G$ be a finite group with $\alpha \in \text{Aut}(G)$ such that $x^{\alpha} \neq x = x^{{\alpha}^2}$ for ...
doctor's user avatar
  • 419
1 vote
0 answers
40 views

Is there any substantial study regarding the automorphism of the field $k((x_1,\cdots, x_n)):=\text{Frac}(k[[x_1, \cdots, x_n]])$?

Let $k$ be a local field, and consider the local ring of the formal power series $k[[x_1, \cdots, x_n]]$. Consider the field of fraction $k((x_1,\cdots, x_n)):=\text{Frac}(K[[x_1, \cdots, x_n]])$. It ...
MAS's user avatar
  • 10.8k
4 votes
1 answer
151 views

Without choice, what can be the (finite) automorphism groups of $\mathbb F_2$-vector spaces?

My motivation for this question is similar to the one in this question. However, that question only asks about the possibility of an infinite $\mathbb F_2$-vector space having trivial automorphism ...
Carla_'s user avatar
  • 41
1 vote
1 answer
88 views

Why must an automorphism of $Z_n$ be of the form $x \mapsto x^a$?

The proof that the automorphism group of $Z_n$ is isomorphic to $(\mathbb Z/n\mathbb Z)^\times$ in Dummit and Foote uses the fact that if $\varphi \in \operatorname{Aut}(Z_n)$ then $\varphi(x) = x^a$ ...
dav's user avatar
  • 121
0 votes
1 answer
43 views

Does a functor from one category to another imply homomorphism between automorphism groups.

I have two categories $C$ and $D$ and a functor $F: C\rightarrow D$. I select an object $A\in ob(C)$, is there guaranteed to be a homomorphism $Aut(A)\rightarrow Aut(F(A))$? It seems like there should ...
FramingCrabbage's user avatar
2 votes
1 answer
51 views

Is an extension of an algebraic group by a multiplicative group a semidirect product?

This is probably a very simple question with a negative answer, but I somehow cannot find a counterexample. Let $X$ be a smooth algebraic variety over an algebraically closed field $k$. Assume that $X$...
L_b's user avatar
  • 698
0 votes
1 answer
69 views

how to calculate the automorphisms of a group that fix a subgroup

I have a finite (polycyclic) group $G$ and a subgroup $H<G$. How do I calculate the subgroup of $Aut(G)$ that fixes $H$ pointwise : $S = \{ a \in Aut(G) : \forall h \in H,a(h)=h\}$ It would be nice ...
unknown's user avatar
  • 1,010
2 votes
1 answer
88 views

What is the outer automorphism group of the Pauli group?

Let $P_n$ be the Pauli group on n qubits. The order of the group is $|P_n| = 2^{2n+2}$. I beleive the outer automorphism group is closely related to the symplectic group $Sp(2n,2)$ but I don't know ...
unknown's user avatar
  • 1,010
0 votes
0 answers
20 views

Automorphism group of a product of two simplices

Is there a description of the group of automorphisms of a product of two simplices depending on the dimension of the simplices? By automorphisms, I mean simplicial automorphisms of a simplicial ...
Grisha Taroyan's user avatar
2 votes
0 answers
54 views

Finding a simple graph such that its automorphism group equals the subgroup of $S_3$ generated by a 3-cycle

I have found that the subgroup of $S_3$ generated by a 3-cycle is $\{e,(123),(132)\}$ where $e$ is the identity but I can't find any graphs that have this group as their automorphism group. I am a ...
mantaray's user avatar
1 vote
0 answers
66 views

When does $\text{Aut}(G)$ act transitively on $n$-tuples of generators of $G$ whose product is 1?

Let $G$ be a finite group. When does the automorphism group $\text{Aut}(G)$ act transitively on a subset of $n$-tuples of generators of $G$ that multiply to 1? I'm interested in this question in ...
utx7563yu's user avatar
3 votes
1 answer
84 views

Automorphism of Labeled Number Line

This is the graph I am working with and the edge labelings are given by $l(\{2n, 2n + 1\}) = a$ and $l(\{2n + 1, 2n + 2\}) = b$. The question I'm working on guides us through to a description of the ...
spooleey's user avatar
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