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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Some questions about set of bijections from $\mathbb{N} \times \mathbb{N}$ to $\mathbb{N}$
Let $S(\mathbb{N}\times\mathbb{N}, \mathbb{N})$ the set of bijective functions from $\mathbb{N}\times\mathbb{N}$ to $\mathbb{N}$. I have some questions about this set.
Note that $S(\mathbb{N}\times\...
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Automorphisms of curves and Hurwitz-Riemann formula
Let $k$ be a base field algebraically closed and of zero characteristic.
Let $C$ be a smooth projective curve and $G$ a finite group of automorphisms of it. Let $C*$ be a smooth projective curve whose ...
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Decomposition of Homomorphism
Is it correct that if $\psi \ : G \rtimes_{\phi_1} H \to G \rtimes_{\phi_2} H$, then $\exists f \in Aut(G)\ and \ g \in Aut(H)$, such that $\psi(G \rtimes_{\phi_1} H) \cong f(G) \times g(H)$
What ...
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Finding automorphisms of a vector space using GAP
Goal:
Given a $K$-dimensional vector space $V \subseteq \mathbb{F}_2^N$ firstly find the group $A$ of all automorphisms $\alpha_i: V\to V $ and secondly find the subgroup $\widetilde{A} \subseteq A$ ...
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How to transform points in subgroup to coset points (quotient group points)?
Elliptic curve parameters: y^2 = x^3 + 3 and prime p = 5119
Let G be the full-group of order 5053.
Let H be the prime-order subgroup (of G) of order = 163.
Let G/H be the quotient group/left coset (...
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Symmetric and asymmetric graphs
Let $\Gamma = (V, E)$ be a self-complementary graph with $|V| \geq 2$, how do i prove that exists $\sigma \in S_{V}$ such that:
I. $xy \in E$ if and only if $\sigma(x)\sigma(y) \notin E$.
II. $\sigma^{...
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Automorphism groups of $\mathbf Z$, $\mathbf Z[i]$, $\mathbf Z[\omega]$
I would like to know the automorphism groups of the rational integers $\mathbf Z$, the Gaussian integers $\mathbf Z[i]$, and the Eisenstein integers $\mathbf Z[\omega]$.
My question is, would $\text{...
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If $G$ is simple, then $Z({\rm Aut}(G)) = \{1\} \iff G$ is non-abelian [duplicate]
NOTE: I am aware of the link at For a Simple Group G, Z(Aut(G)) Is Trivial if and only if G is Non-Abelian. I am looking for proof verification of $G$ non-abelian $\implies Z(\mathrm{Aut}(G)) = \{1\}$....
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Questions about *-automorphism and C*-algebras
My doubts are probably very trivial, but I am not 100% sure about that, so any help would be much appreciated. Let's suppose we have a C*-algebra $\mathcal{A}$ and $\tau$ is a *-automorphism. Let's ...
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Regular permutation groups and 2-closure
I'm currently working through Dobson, Malnič, Marušič's Symmetry in Graphs (2022), and am trying to prove the following proposition (pp. 208):
Exercise 5.2.1 If $G \leqslant \text{Sym}(\Omega)$ is ...
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Automorphism group of Cayley graph much larger than the subgroup preserving directedness
Let $G$ be a group with generating set $S$ and $\Gamma$ the corresponding Cayley graph. Let $Aut(\Gamma)$ be the automorphism group of the corresponding undirected graph and $Aut^+(\Gamma)$ the ...
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When the natural homomorphism from $Aut(G)$ to $Aut(G/N)$ is onto?
For the last few days, I have been working on a bunch of group theory problems and I am stuck with one question that came into my mind. My level is up to an introductory course in group theory and ...
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Show that $T$ is a topological embedding of $X$ in itself.
Let $X$ be the set of odd positive fractions with denominator $3$ in lowest terms, except for $\frac13$.
$X=\{x\in\Bbb Q^+:3x\in3\Bbb N\pm1\}\setminus\{\frac13\}$
If $a,b$ are 2-adic units, then any 2-...
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What is the sum of the 2-adic values of the orbits of the affine group over 2-adic integers restricted to $ax+b$ where $2^{\nu_2(x)}=2^{\nu_2(b)}$?
Let $\textrm{aff}(\Bbb Z_2)$ be the affine group over 2-adic integers, defined as the linear polynomials $ax+b$ where $a\in\Bbb Z_2^\times$ and $b\in\Bbb Z_2$.
A very interesting subset of this group ...
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Automorphism group cyclic implies abelian group, do we have more?
I am working on this exercise in Lang's Algebra:
Exercise I.7: Let $G$ be a group such that $\text{Aut}(G)$ is cyclic. Prove that $G$ is abelian
I have shown the set of inner-automorphisms form a (...
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Automorphism group of a Polytope
I have been given the following task: Let $P \subseteq \mathbb{R}^d$, a polytope, such that $P = \text{conv}(K)$ for $K \subseteq S^{d-1}$ finite, where $S^{d-1}$ denotes the Euclidean unit sphere in $...
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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 $G$-...
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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 for direct products of finite commutative nilpotent rings.
Let $(R, +, \cdot)$ be an associative commutative nilpotent ring of cardinality $2^n$ such that
$$
r^2 = 0,
$$
for every $r\in R$. Also $(V, +)$ is a vector space over $\mathbb{F}_2$. Let $\...
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About using onto and into in the definition of automorphism
Definition. An isomorphism $\phi: G \to G$ of a group $G$ with itself is an automorphism of G.
Why is it that rephrasing this definition with "An automorphism of a group G is an isomorphism ...
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Automorphism of unit disc which maps two points on the boundary to $1$ and $-1$
Let $\mathbb{D}=\{ z \in \mathbb{C} : |z|<1\}$ and its boundary $\mathbb{T}=\{ z \in \mathbb{C} : |z|=1\}$. If $\lambda \ne \mu$ are two points in $\mathbb{T}$, then I need to show that there is an ...
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If $\langle f_1,f_2 \rangle = \langle g_1,g_2 \rangle$, then $\langle f_1-\lambda,f_2-\mu \rangle = ?$, in terms of $g_1,g_2$
Let $f_1,f_2,g_1,g_2 \in \mathbb{C}[x,y]-\mathbb{C}$.
Assume that $\langle f_1,f_2 \rangle = \langle g_1,g_2 \rangle \subsetneq \mathbb{C}[x,y]$.
Claim:
Given $\lambda,\mu \in \mathbb{C}$, $(\lambda,\...
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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 ...
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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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Can we give the complete list of $n$ such that there exists a group of order $n$ whose automorphism group is nontrivial and of odd order?
I was wondering if it is possible to determine all $n$ such that there exists a group $G$ with odd $|\operatorname{Aut}(G)|>1$. The following numbers are some examples that occur in the list:
$729=...
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Finding orbit in a group action of automorphism group of dihedral group on the dihedral group.
Finding orbit in a group action of automorphism group of dihedral group on the dihedral group.
Let $$D_{2n}=\langle a,b:a^n=b^2=1,bab^{-1}=a^{-1}\rangle $$ be the dihedral group of order $2n$ and $...
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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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A cyclic group of order 25 doesn't have an automorphism of order 3.
Let $G$ a cyclic group of order 25. Prove $\forall \phi \in{\rm Aut}(G):|\phi| \neq 3$.
I am a bit stuck and I'm not sure I'm even going in the right path.
Assume with contradiction $\exists \phi \in ...
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Compute the automorphism group of $D_{10}$ [duplicate]
This exercise will compute the automorphism group of $D_{10}$. The presentation of the group
is $D_{10 }= <r, s |r^5 = s^2 = 1, s^{−1}rs = r^{−1}>$
(a) (i) Show that under any automorphism $σ : ...
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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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Size of $p$-subgroups of $\operatorname{Aut}(G)$, where $p$ divides the order of $G$
Let $G$ be a nontrivial finite group. In his book "Finite Group Theory", M. Isaacs proves the following two results:
Corollary 3.3: Let $\sigma \in \operatorname{Aut}(G)$. Then, $o(\sigma) &...
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Existence of graph automorphism with certain property, without enumerating the automorphism group.
Let $G$ be a directed graph, $A, B \subseteq V(G)$ be two subsets of its vertices, and $S$ be a generating set of its automorphism group $\operatorname{Aut}(G)$.
How can we determine whether there is ...
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Computing $\mathrm{Fix}(\phi)$ for autormophisms $\phi$ of free groups
Let $F_A$ be the free group generated by the finite set $A$ and let $\phi\colon F_A \to F_A$ be a group-automorphism. It is known [1] that
$$ \mathrm{Fix}(\phi) = \{g \in F_A : \phi(g) = g\} $$
is (...
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A "simpler" description of the automorphism group of the Lamplighter group
The lamplighter group is defined by the following presentation:
$$
L_N=(\mathbb{Z} _N) \wr \mathbb{Z} \cong\left\langle t, a_n, n \in \mathbb{Z} \mid a_n^N, t a_n t^{-1}=a_{n+1}, n \in \mathbb{Z}, a_n ...
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What is ${\rm Aut}\bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_p$?
Let $p$ be a prime, what is the automorphism group of $G=\bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_p$ (countable infinite direct sum of $\mathbb{Z}_p$)?
I know every permutation of the coordinates will ...
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Prove $Aut(B_{/A})$ of $B$ inducing the identity on $A$ is finite if and only if the group of invertible elements of $A$ is finite.
Let $A$ be a domain and $B = A[T, \frac1T]$ for an indeterminate $T$. Prove that the ring automorphism group $Aut(B_{/A})$ of $B$ inducing the identity on $A$ is finite if and only if the group of ...
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When is a group acting on an algebraic variety $X$ a normal subgroup of $Aut(X)$?
Let $G$ be a group acting faithfully and algebraically on an algebraic variety $X$. Then we can understand $G$ as a subgroup of the automorphisms group $Aut(X)$ of $X$. My question is: when is $G$ a ...
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Let $\mathfrak{A}$ be a finite dimensional $k$-algebra for alg. closed $k$. Prove ${\rm Aut}(\mathfrak{A})$ is a closed subgroup of $GL(\mathfrak{A})$
This is Exercise 7.6.3 of Humphreys', "Linear Algebraic Groups".
The Question:
Let $\mathfrak{A}$ be a finite dimensional $k$-algebra for algebraically closed field $k$. Prove that ${\rm ...
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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 ...
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Automorphism group of $C_{4} \times C_{2}$
Let $C_{4} = \langle a \rangle$ a cyclic group of order 4, denoted multiplicatively. Let $C_{2} = \langle b \rangle$ a group of order 2, also denoted multiplicatively.
The automorphism group of $C_{4} ...
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How many automorphisms are in $Z_2 \times Z_2 \times Z_2$?
I came across this question on brilliant. I first observed that any particular automorphism (i.e., a bijective homomorphism) must map each group element to a distinct element in the group set: if it ...
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