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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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Order of automorphism group of cyclic group

Let $G$ be a cyclic group of order $m$. What is the order of $\text{Aut}(G)$? I want to know the proof as well (elementary if possible). I would still accept the proof if one answers with $m = p$, a ...
21
votes
1answer
208 views

On automorphisms of groups which extend as automorphisms to every larger group

For a group $G$, let $\operatorname{Aut}(G)$ denote the group of all automorphisms of $G$ and $\operatorname{Inn}(G)$ denote the subgroup of all autmorphisms which is of the form $f_h(g)=hgh^{-1}, \...
12
votes
0answers
235 views

Are all verbal automorphisms inner power automorphisms?

Suppose $G$ is a group. $\DeclareMathOperator{\Wa}{Wa}\DeclareMathOperator{\Tame}{Tame}\DeclareMathOperator{\Aut}{Aut}$ Lets call $\phi \in \Aut(G)$ verbal automorphism iff $\exists n \in \mathbb{N}, ...
11
votes
2answers
177 views

Determining an inner automorphism via queries

Let $G$ be a group. You are given that $\phi$ is some inner automorphism of $G$, i.e, $\phi(x) = axa^{-1}$ for some $a \in G$ uniquely determined modulo the center of the group. You're not sure what $...
11
votes
1answer
153 views

Deck transformation group in algebraic geometry

Let $f:X\to Y$ be a finite morphism between (irreducible) varieties. We can define $\operatorname{Aut}(X/Y)$ to be the automorphism of $X$ commutes with $f$. For the case over $\mathbb C$, we can also ...
10
votes
2answers
170 views

Does there exist a finite group whose automorphism group is simple?

Ignore any trivial cases. Also, even if some examples exist for $|G|$ small, I still want to know if there are simple automorphism groups for arbitrarily large $|G|$. For such a group to exist, ...
10
votes
1answer
296 views

Extension of isomorphism of fields

I'm reading the book of Razmyslov "Identities of Algebras and their representations" and he uses some "supposedly known fact" from field theory. As I could understand, a more or less general statement ...
10
votes
1answer
60 views

How large can the outer automorphism group be?

You can make the outer automorphism group very large by taking $G$ be to be a vector space (say over a finite field) so that if $|G| = q^n$, then $Out(G) = GL_n(\mathbb F_q)$ of size exponential in $n$...
9
votes
1answer
76 views

If $f$ has no non trivial fixed points and $f\circ f$ is the identity then $f(x)=x^{-1}$ and $G$ is abelian for $f$ an automorphism of $G$ [duplicate]

Let $f$ be an automorphism of the finite group $G$ such that $f\circ f=id$ and $f(x)=x\implies x=e$ Prove that $f(x)=x^{-1}~\forall x\in G$ If we can prove that $f(x)$ and $x$ commute for any ...
8
votes
2answers
85 views

Group isomorphism $h:(\mathbb R,+)\to (\mathbb R^+,\times)$ that is not an exponential function.

Let $\mathbb{R}^+$ denote the set of positive real numbers. Group isomorphism $h_b:(\mathbb R,+)\to (\mathbb R^+,\times)$ can be given by the exponential function: $h_b(r)=b^r$, where $b$ is a ...
8
votes
2answers
157 views

Is there a way to describe all finite groups $G$ such that $\operatorname{Aut}(G) \cong S_3$?

Is there a way to describe all finite groups $G$ such that $\operatorname{Aut}(G) \cong S_3$? Two groups that definitely satisfy that condition are $S_3$ itself (as it is a complete group) and $\...
8
votes
1answer
65 views

If a function has an infinite amount of automorphisms would that imply that it is periodic?

Let $\phi(x)$ be continuous and differentiable everywhere and its automorphisms be bijections $\gamma_{n}$ such that $\phi(\gamma_{n}(x))=\phi(x)$. If the order of $Aut(\phi)$ is infinite, would that ...
8
votes
2answers
142 views

Automorphism group of the projective unitary group PU(N) and SO(N)

I would like to determine the automorphism group of the projective unitary group $G=PU(N)=PSU(N)$ and $G=SO(N)$. We also knew that $$ 0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0. $$...
8
votes
1answer
118 views

Variety with nonalgebraic automorphism group

In Algebraic Geometry - A First Course by Joe Harris, an example in section 10 on algebraic groups reads that there exist varieties whose automorphism group is not an algebraic group. To give an ...
8
votes
0answers
69 views

What can we say about the equivalence relation defined such that $G \sim H \Longleftrightarrow Aut(G) \simeq Aut(H)$?

Given two finite graphs $G, H$, I say that $G \sim H$ if and only if $Aut(G) \simeq Aut(H)$. We define $[G] = \{H \mid Aut(G) \simeq Aut(H) \}$ for any graph $G$. Due to theorems by Erdos and others ...
7
votes
1answer
141 views

Groups which can not occur as automorphism group of a group

Consider the following natural question: Given a finite group $H$, does there exists a finite group $G$ such that $\mathrm{Aut}(G)\cong H$? In short, does any finite group occurs as the ...
7
votes
1answer
136 views

What is an example of a power automorphism of a group that isn't a universal power automorphism?

A power automorphism maps every subgroup of a group to within itself, with equality if the group is finite. More specifically, for a subgroup $H$ of $G$, a power automorphism $f$ has $f(H) \subseteq H$...
7
votes
0answers
30 views

What is the smallest poset with automorphism group $C_n$?

I've recently been interested in finding small finite posets (and thereby finite $T_0$ topologies) with a given automorphism group. I came upon the paper of Barmak and Minian in which they provide an ...
6
votes
2answers
3k views

A characteristic subgroup is a normal subgroup

$\phi(H) = H$ for $\phi$ any automorphism in $G$. I tried to find a homomorphism for which $H$ is the kernel, which shows that $H$ is normal. However, I tried to have it maps $g$ to $\phi(gH) = \phi(...
6
votes
1answer
97 views

Does every finite non-trivial complete group have even order?

Does every finite non-trivial complete group have even order? I checked three well known classes of complete groups, and this statement is true for them all: 1) Symmetric groups: All symmetric ...
6
votes
2answers
137 views

Is it true that if $\mathrm{Aut}(G)$ is nilpotent then $G$ is also nilpotent?

Is it true that if automorphism group of $G$ is nilpotent then $G$ is also nilpotent?
6
votes
1answer
124 views

Are all power automorphisms of a finite abelian group of the form $f(x)=x^n$?

An example of a universal power automorphism, that is automorphisms of the form $f(x)=x^n$, is easy to find in an abelian group. Any power $n$ will work as long it is relatively prime to the orders of ...
6
votes
1answer
54 views

Question about Isomorphism of Aut(G)

In material supplied by my instructor there is a question which asks to pick the incorrect statement among the following: If $\text{Aut}(G_1)\cong \text{Aut}(G_2)$ and $G_1$ is infinite group then $...
5
votes
1answer
50 views

Appeal for clarification of an isomorphism between $\operatorname{Aut}_c(G)$ and $\operatorname{Hom}(G,Z(G))$

I am reading an older paper by Jamali and Mousavi. On the second page there is the following proposition 2.2 I marked fourplaces in red. The first one seems like a typo: ".. for every $f$ in $\...
5
votes
1answer
338 views

Group isomorphic to its automorphism group

Any complete group (that is, with trivial center and outer automorphism group) is isomorphic to its automorphism group. The inverse is not true, as the dihedral group of order 8 is isomorphic to its ...
5
votes
1answer
130 views

Is the statement that $ \operatorname{Aut}( \operatorname{Hol}(Z_n)) \cong \operatorname{Hol}(Z_n)$ true for every odd $n$?

Is the statement that $ \operatorname{Aut}(\operatorname{Hol}(Z_n)) \cong \operatorname{Hol}(Z_n)$ true for every odd $n$? $Hol$ stands here for group holomorph. This problem appeared, when I ...
5
votes
1answer
96 views

The automorphism group of $S_6$ is isomorphic to a semidirect prodct

On this document an outer automorphism of $S_6$ is constructed. I would like to use this construction to prove that $\mathrm{Aut}(S_6)\cong S_6\rtimes_\varphi\mathbb{Z}_2$. The idea would be to find ...
5
votes
0answers
64 views

Automorphism group of a finite group [closed]

For which finite groups $G$ is $\operatorname{Aut}(G)$ isomorphic to a (non necessarily strict) subgroup of $G$?
4
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2answers
99 views

Hölder's theorem (Group Theory). Proof collection and applications

In (finite) group theory, the number $6$ seems to be special in the sense that we have the result of Hölder that this is the only natural number $n=6$ such that there exists an outer automorphism of $...
4
votes
2answers
52 views

Examples of a group $G$ with a non-trivial homomorphism $f:G \to Z(G)$

I recently learned that if $f: G \to Z(G)$ is a homomorphism of $G$ to its center, then $g:G \to G$ defined as $g(x)=f(x)x$ is an endomorphism of $G$. I am having trouble thinking of examples of a (...
4
votes
2answers
135 views

Gordon Royle's 21-vertex 21-automorphism graph

OEIS A080803 lists the minimal number of vertices $a(n)$ needed to support an undirected graph whose automorphism group has order $n$. The MathWorld page on graph automorphisms links to this sequence ...
4
votes
1answer
72 views

Is there a way to describe the structure of $Aut(UT(3, p))$?

Is there a way to describe the structure of the automorphism group of $$C_{p}^2 \rtimes C_p \cong \langle x, y, z | [x,y]=z, [x,z]=[y,z]=x^p=y^p=z^p=e \rangle \cong UT(3, p)?$$ Here $p$ is an odd ...
4
votes
1answer
139 views

Is there a way to classify all metabelian finite groups $G$, such that $ \operatorname{Aut}(G) \cong G$?

Is there a way to classify all metabelian finite groups $G$, such that $ \operatorname{Aut}(G) \cong G$? I know that the trivial group is the only abelian group that satisfies this condition. I also ...
4
votes
2answers
15 views

Subgroup of coprime order with automorphism group is contained in center of group

I'm studying for a qualifying exam in algebra and I've come across the following problem: Let $G$ be a finite group with a subgroup $N$. Let $Aut(G)$ be the group of automorphisms of $G$. Prove ...
4
votes
1answer
42 views

The Automorphism Group of Free Quandles?

The following definitions are quoted from this article: My question is, do we know anything description the automorphism group of free quandles with relative smaller number of generating set $S$? For ...
4
votes
1answer
95 views

Determine invariant subfield and order in $\operatorname{Aut}(L)$

I'm trying to determine some explicit invariant subfields of $L=\mathbb{Q}(X)$. We have defined $\sigma_i\in$ Aut$(L)$ with $$\sigma_1(X)=1/X,\ \sigma_2(X)=1-X.$$ We are now asked to determine the ...
4
votes
1answer
71 views

Automorphism group of ($\mathbb{R}^{\times}, \cdot$)

Let ($\mathbb{R}^{\times}, \cdot$) be the multiplicative group of non-zero real numbers. I want to find the automorphism group of ($\mathbb{R}^{\times}, \cdot$). My guess is that it is the group of ...
4
votes
1answer
110 views

What are non-canonical generators of $\hat{\mathbb{Z}}$ (resp. the absolute Galois group of a finite field)?

We define $\hat{\mathbb{Z}}$ as the inverse limit $\varprojlim \mathbb{Z}/n \mathbb{Z}$. One can also show that $\hat{\mathbb{Z}}$ is isomorphic to the absolute Galois group $G_k = \operatorname{Gal}(\...
4
votes
1answer
85 views

Why we can represent automorphisms in $\text{Gal}(\Bbb Q(\sqrt[4]{2},i)/\Bbb Q)$ as permutations in $S_4$ but not $S_8$?

The splitting field of $x^4-2$ over $\Bbb Q$ is $G=\text{Gal}(\Bbb Q(\sqrt[4]{2},i)/\Bbb Q)$. By primitive element theorem, $K=\Bbb Q(\alpha)$ for some $\alpha$ and $[K:\Bbb Q]=8$. So I know that the ...
4
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0answers
59 views

Example of when $E^{H} \neq E^{\text{Aut}(E/E^H)}$?

Let $E/F$ be a field extension of $F$, and let $H \leq \text{Aut}(E/F)$ be a subgroup of the automorphisms of $E$ fixing $F$. If $H$ is finite, then it is well-known that $H = \text{Aut}(E/E^H)$ and ...
4
votes
0answers
113 views

For any sets $X$ & $Y$ can the group of automorphisms for the digraph $G(X,Y)=(X\cup Y,X\times Y)$ be express in terms of symmetric groups?

For any sets $X,Y\neq \emptyset$ if we define a permutation group $G(X,Y)$ on the set $X\cup Y$ as follows: $$G(X,Y)=\left\{\sigma\in \text{Sym}(X\cup Y):\forall x,y\left[(x,y)\in X\times Y\iff (\...
4
votes
0answers
109 views

Automorphisms of $\mathbb{A}^1_R$

When $R$ is an integral domain, the automorphisms of the affine line are all of the form $X \mapsto aX + b$ with $a \in R^\times$ and $b \in R$; the proof is the same as in the case of $R$ a field, ...
3
votes
4answers
187 views

Automorphism in $U(16)$.

Here $U(16)$ is set of integers less than 16 that are coprime to it. We have to prove that mapping $f:x\to x^3$ is an automorphism. Here I am not able to prove that $f$ is onto. Is there a general ...
3
votes
3answers
235 views

Why Automorphisms in Galois theory (why not Homomorphisms to itself)?

I am wondering, why are Automorphism needed in Galois theory, why is it not enough that we have group homomorhisms $f$, that obey $$f(a+b)=f(a)+f(b)$$ and $$f(a\cdot b)=f(a)\cdot f(b)$$ where both $x$ ...
3
votes
2answers
91 views

Find the order of $\operatorname{Aut}(G)$.

Let $G$ be a non-abelian group of order $20$. What will be the order of $\operatorname{Aut}(G)$? $(a)$ $1$. $(b)$ $10$. $(c)$ $30$. $(d)$ $40$. If I take $G = D_{10}$ then I find that the order ...
3
votes
2answers
67 views

Why are there only two groups of order 2p for $p=5,7$?

It is clear to me that using semidirect products we can get the groups of order $2p$ for $p=5,7$. I know that there is only one order 2 automorphism of $\mathbb{Z}_q$ for $p=5,7$. How can I use this ...
3
votes
2answers
78 views

Constructing a polynomial given the Galois Group of it's splitting field

Let $red_p : \mathbb{Z}[x]\to\mathbb{Z}/(p)[x]$ be the canonical ring morphism sending a polynomial with integer coefficients to a polynomial with integer coefficients modulo $p$, with $p$ a prime, by ...
3
votes
1answer
70 views

Question of Automorphism $T$ on Finite group with property that $T(x)=x$ only for $x=e$

Let $G$ be finite group, $T$ be automorphism on $G$ with property that $T(x)=x$ only for $x=e$. Then 1) every $g \in G $ can be written as $g=T(x)x^{-1}$ for some $x\in G$ 2) Furthermore if $T^2=\...
3
votes
1answer
43 views

Holomorph of a group $G$, then the automorphism of $G$ are inner automorphisms

In my course notes of algebra it says: Let $G$ be a group. Then $\mathrm{Aut}(G)$ acts on $G$ in a natural way through automorphisms. This allows us to consider $A:= G \rtimes \mathrm{Aut}(G)$. In ...
3
votes
1answer
146 views

Boundary $\partial F_n$ of a free group $F_n$

I am reading "The Tits alternative for $\operatorname{Out}(F_n)$ I: Dynamics of exponentially-growing automorphisms" by Mladen Bestvina et al. for my Master's thesis. On page 526, the following part ...