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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Automorphism of commutative groups.

For every group G there is a natural group homomorphism G → Aut(G) whose image is the group Inn(G) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can ...
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Automorphism group of a cuspidal “elliptic” curve

I have a cuspidal elliptic curve ($\Delta=0$ and $J$-invariant$=0$), $E$ in a field $K$ of characteristic $3$ and I'm trying to show that its automorphism group is $K^*$. These are my calculations: $E$...
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Verify question on automorphism

The question is as follows in a past exam paper: Suppose that $G$ is a group. An isomorphism from $G$ to itself is called an automorphism. Prove that the set $Aut(G)$ of all automorphisms of $G$ is ...
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Automorphism group of elliptic curves

I would show that the automorphism group of an elliptic curve E in characteristic $3$ and with J-invariant $0$ is isomorphic to the semidirect product of $Z/4Z$ and $Z/3Z$. The curve has equation of ...
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Automorphism group of rooted tree is a profinite group

I'm working with the book Self-similar Groups by Volodymyr Nekrashevych. In the chapter $1$ he statements the following proposition: We have an equality $St_{Aut(X^*)}(n) = RiSt_{Aut(X^*)}(n)$. The ...
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Finding $|\!\operatorname{Aut}(L(K_4))|$ using Orbit-Stabiliser Theorem

I know that you can find the size of an automorphism group of a simple graph $G$ by using the Orbit-Stabiliser theorem as follows: let $\DeclareMathOperator{Aut}{Aut}A = \Aut(G)$, and $v$ be a vertex ...
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Describe the automorphism group of $Aut(\mathbb{Z}/9\mathbb{Z})$.

a)Describe the automorphism group of $Aut(\mathbb{Z}/9\mathbb{Z})$. b) Prove that if a Group G has the trivial center then $|Aut(G)| \geq |G|$. My attempt: a) Clearly, $\mathbb{Z}/9\mathbb{Z}$ is a ...
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Is Aut(G×H) isomorphic to Aut(G) × Aut(H) [duplicate]

Please suggest me the proof.Am stuck with it.I saw somewhere that it will be true if (o(G),o(H))=1 but why?
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Q about torus automorphism example

I am working through the book "introduction to tropical algebra" and page 70 and 71 example 2.2.10 states "An automorphism of the torus is an invertible map specified by Laurent monomials. Thus the ...
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$G/F(G)$ is isomorphic to $X_1\times\cdots\times X_t$

Since I still don’t know the answer, I’ve also asked it on math.overflow. I saw a remark in an old post that $G/F(G)$ is isomorphic to a group of the form $X_1 \times \ldots \times X_t,$ where ...
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Conditions imposed on graph automorphism leave the “voltage” on graph nodes identical?

We call $\pi$ a graph automorphism on a graph $(V,E)$ if $\pi:V\to V$ is a bijection such that $xy\in E\iff \pi(x)\pi(y)$ is an edge, and $\phi(xy)=\phi(\pi(x)\pi(y))$, where $\phi$ is the conductance....
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About the order of an automorphism

Let $G=\mathbb{Z}_{p^n}\times \ldots\times\mathbb{Z}_{p^n}$ be a direct sum of $m>1$ cyclic groups of order $p^n$. Suppose you have an automorphism $\alpha$ acting irreducibly on $\Omega_1(G)$, on $...
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If $[G : H]$ and $|H|$ are coprime, than $H \operatorname{char} G$ [duplicate]

Let $G$ be a finite group, and let $H⊲G$ so that $|H|$ and $[G : H]$ are coprime. Prove or disprove that $H$ is a characteristic subgroup of $G$, that is, for every automorphism $\varphi\in \...
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Showing that there exists an automorphism of a cyclic group mapping a generator to another.

I'm currently going through Lang's Algebra, I've only recently started it, and I'm trying to do every proof left as an exercise. One such question is the following : Let $G$ be a cyclic group, and ...
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Showing that $Aut(S_n)=S_n$ for $n>6$ with a argument of centralizers.

In this exercise we shall prove that $Aut(S_n) =S_n$ for $n > 6$. (The results holds true for $n = 4, 5$ too and fails for $n = 6$.) Thus, $S_n$ is complete for $n > 6$. (a) Prove that an ...
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Permutations of a subset of combinations

Let $C$ be some subset of all the $ \binom{n}{k}$ combinations of $k$ out of $n$ elements. What is the group of permutations $\pi: C \to C$ of the indices of the combinations mapping $C$ to itself, i....
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The direct product of quotients is a quotient of the direct product

I asked a question about if we have in general that if $G=G_1\times \cdots \times G_n$ (where $G_i$ are characteristic in $G$ for $i=1,\cdots ,n$), then $${\rm Out}(G)\cong {\rm Out}(G_1)\times\cdots\...
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How do automorphisms of a group act on the set of irreducible representations?

Let $G$ be a finite group and $Rep(G)$ denote the set of all the irreducuble representations of $G$ (up to isomorphism, of course, so it is a finite set). The problem is to define an action of $Aut(...
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Do we have ${\rm Out}(G)\cong {\rm Out}(G_1)\times\cdots\times {\rm Out}(G_n)$?

Linked to this question I know that if $G_i$ are characteristic subgroups of $G$ (for $i=1,\cdots ,n$) and $G=G_1\times \cdots\times G_n$, then ${\rm Aut}(G)\cong {\rm Aut}(G_1)\times\cdots\times {\...
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Frattini subgroup of $p$-groups and the automorphisms

About $p$-groups, I saw on Wikipedia that: Every automorphism of $G$ induces an automorphism on $G/Φ(G)$, where $Φ(G)$ is the Frattini subgroup of $G$. The quotient $G/Φ(G)$ is an elementary ...
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Why does every $\varphi: K \to \mathrm{Out}(H)$ determine an unique extension of $H$ by $K$ when $Z(H) = 1$?

Every homomorphism $\varphi: K \to \mathrm{Out}(H)$ determines an unique extension of $H$ by $K$. Why is this true for groups $H$ with a trivial center? Even if we only consider split extensions, as ...
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The automorphism of splitting field of x^p-x+a over Z_p.

I'd like to solve the question. Let $L$ the splitting field of $f(x)=x^p-x+a$ ($a\neq 0$) over $\mathbb Z_p$. $g : L \rightarrow L$, $g(\alpha)=\alpha+1$ where $\alpha$ is a root of $f(x)$ is ...
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Why $G/F(G)$ is isomorphic to a subgroup of $\mathrm{Out}(F(G))$?

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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Is $\mathrm{Aut}$ a functor/invariant? [duplicate]

Can the following assignment on objects be made into a functor from the category of topological spaces into the category of groups? Each topological space $X$ gets mapped to $\mathrm{Aut}(X)$, its ...
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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 confusing. Let $G$ be a finite solvable group and $F(G)$ is the Fitting subgroup of $G$. (1) $G/Z(...
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The Galois group ${\rm Gal}(\Bbb C/\Bbb Q) ({\rm Aut}(\Bbb C/\Bbb Q)).$

I solved at For $f$ in ${\rm Gal}(\Bbb C/\Bbb Q),$ $F(i)$ is $i$ or $-i$ And for $a,b$ in $\Bbb Q$, $f(a+bi)=a+bf(i)$. I have to classify $G(\Bbb C/\Bbb Q)$, but I find the automorphism $f$ is ...
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Let $|G|=n,m\in\Bbb Z$ with $\gcd(m,n)=1$. Let $\sigma:G\to G$ be $\sigma(g)=g^{m}\forall g\in G$. If $G$ is abelian, then $\sigma\in{\rm Aut}(G)$.

Here is the problem. Let $|G|=n,$ and let $m$ be an integer with $\operatorname{gcd}(m, n)=1 .$ Define $\sigma: G \rightarrow G$ by $\sigma(g)=g^{m}$ for all $g \in G .$ If $G$ is abelian, show ...
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Automorphism of finite extensions $\mathbb{Q} \subset K$

Let $\mathbb{Q} \subset K$ be a finite field extension and $\varphi \in \text{Emb}(K,K)$. I want to show that if $\varphi(x)=y$ then $\varphi (\overline{x}) = \overline{y}$, where $\overline{x}$ ...
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Automorphism group of $F$, where $F$ is the quotient field of the integral domain $R=\Bbb Z[x]/(x^3+x+1)$

Consider the ring $R=\Bbb Z[x]/(x^3+x+1)$. It is easily seen that $(x^3+x+1)$ is a prime ideal, so that $R$ is an integral domain. Consider its field of fractions $F$. I am asked to describe the group ...
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Prove $\text{Inn}(\mathbb{Q_8}) \cong \mathbb{Q_8}/ Z(\mathbb{Q_8})$ [closed]

I found on wikipedia somewhere that $\text{Inn}(\mathbb{Q_8}) \cong \mathbb{Q_8}/ Z(\mathbb{Q_8})$ but I don't know how to prove that their orders are equal. Also, note: $Z(\mathbb{Q_8}) = \{-1,1\}$ ...
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Equivalent definitions of $C^*$-systems.

I'm trying to compare the two following definitions of $C^*$-system trying to translate the one into the other. Definition 1: A $C^*-$dynamical system is a triple $(\mathfrak{A},G,\alpha)$ where $\...
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Let $G$ be group and $f:G\rightarrow G$ be automorphism. If $Z$ be centralizer group, prove that $f(z)=Z$.

Here's the complete problem. Let $G$ be group and $f:G\rightarrow G$ be automorphism. Define a set $Z=\lbrace z \in G | zg=gz, \forall g \in G \rbrace$ and $f(Z)=\lbrace F(z)|z \in Z \rbrace.$ Prove ...
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Calculation of the fixed field of a subgroup of a Galois group

Let $L/K$ be a Galois extension. I would like to understand how to compute the fixed field of a subgroup $H \leq Gal(L/K)$ as explicitly as possible. The Fundamental theorem of Galois theory often ...
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Number of homomorphism from $\mathbb Z_5 \to S_6$

I am finding the number of homomorphisms from $\mathbb Z_5 \to S_6$. We see the only subgroups of $\mathbb Z_5$ are $\mathbb Z_5$ and ${0}$. $\frac{\mathbb Z_5}{\mathbb Z_5}$ is isomorphic to onle ...
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If $L/K$ is a field extension, is $\text{Aut}(K)$ a normal subgroup of $\text{Aut}(L)$?

If $L/K$ is a field extension, is $\text{Aut}(K)$ a normal subgroup of $\text{Aut}(L)$ (possibly under some extra conditions on $L$ and $K$) and if so what is the quotient isomorphic to, possibly $\...
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Automorphism group of poset of models of Peano Arithmetic.

Suppose we put a poset structure on the set of countable models of Peano Arithmetic as follows: for models 𝑃 and 𝑄, let 𝑃≤𝑄 if 𝑃 is isomorphic to a submodel of 𝑄. As described, this is just a ...
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Marcus Ch4 Problem 9

Let $L$ be a normal extension of $K, P$ a prime of $K, Q$ and $Q^{\prime}$ primes of $L$ lying over $P .$ We know $Q^{\prime}=\sigma Q$ for some $\sigma \in G .$ Let $D$ and $E$ be the decomposition ...
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Consider the map $\psi : G \rightarrow Aut(G)$ given by $\psi(g) = \phi_{g}.$ Prove that $\psi$ is a homomorphism [duplicate]

In a previous problem, I already proved that $\phi_{g}$ is an isomorphism, for $\phi_{g}(x)=gxg^{-1}$ so knowing that $\phi_{g} = gxg^{-1}$ is an isomorphism will certainly help here. Anyway, this is ...
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Understanding Automorphism in Galois for Beginners by John Stillwell

I am having trouble to understand automorphism in the article "Galois for beginners" by John Stillwell, like my friend Leo's post and Alberto's post. The main question is related to the automorphism ...
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Image of a finite set in $\mathbb{P}^1$ under automorphism

I'm having a hard time trying to prove the following statement : Prove that for finite sets $\{ P_1, ... , P_m \}$ and $\{ Q_1, ... , Q_n \}$ of $\mathbb{P}^1$, there exists $ A \in GL(2,\mathbb{C}) ...
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Automorphism of $\mathcal{M}_n(\mathbb{C})^r$

I'm stucked with this question, I have no clue. We denote $\mathcal{M}_n(\mathbb{C})^r:=\mathcal{M}_n(\mathbb{C})\times\ldots\times\mathcal{M}_n(\mathbb{C})$ $r$ times, I have to show that if $f$ is ...
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Finding automorphism group for finite fields

There is a question about Galois Theory I found: "Let $L$ be a field and $K$ its prime subfield. Let $\phi$ be an automorphism of $L$. Show that $\phi$ is an automorphism of $L/K$" All I need is to ...
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A question in proof of a theorem in Galois Theory from Hungerford Algebra

I am self studying Fields and Galois Theory from Algebra by Thomas Hungerford and I have a question in this theorem s proof. Its image I have question in line 1 where author writes "In any case $...
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How is Aut(G) = Aut(G¯)? (where G¯ is the complement of G)

For a Graph G, I am trying to understand how the automorphism of G is equivalent to the automorphism of the complement of G. I understand that an automorphism is an isomorphism of G onto itself. This,...
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Categories where possible automorphism groups are not understood.

In many categories, such as the category of graphs or topological spaces, every group appears as an automorphism group of an object in that category. This certainly isn't true for all categories, even ...
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Automorphism group of finite field extension has a trivial stabilizer

We have this theorem. Let $L|K$ a field extension with $[L:K]<\infty$ and $G=\text{Aut}(L|K)$. We let $G$ act on $L$. Then there is a trivial stabilizer. The proof is the following, I would ...
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Exercise 0.6ii in Miles Reid's Commutative Algebra (proof verification)

Let $K=k(T)$ be the field of rational functions; a k-automorphism of $K$ is a ring homomorphism $\phi: K\rightarrow K$ that is the identity on $k$ and is an automorphism of $K$. Describe the group $\...
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Must large (infinite) groups have large automorphism groups?

For every cardinal $\kappa$, is there a cardinal $\lambda$ such that for all groups $G$ with $|G| > \lambda$, we have $|\mathrm{Aut}(G)| > \kappa$? I believe a similar result holds for finite ...
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Suppose $(n,m)=1$. Prove any automorphism of $\mathbb{Z}_n \times \mathbb{Z}_m$ is of the form $\varphi(x, y) = (ax, by)$.

Suppose $n, m$ are coprime. I need to show that any automorphism $\varphi : \mathbb{Z}_n \times \mathbb{Z}_m$ is of the form $\varphi(x, y) = (ax, by)$ for some $a, b$ with $(a,n)=(b,m)=1$. I'm ...

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