The study of symmetry: groups, subgroups, homomorphisms, group actions.

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Classification of the decomposable primitive permutation groups

It is seen in comments here that the diagonal subgroup of the finite group $G \times G$ is core-free maximal iff $G$ is a nonabelian simple group. This gives examples of decomposable primitive ...
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227 views

Galois and solvable primitive permutation groups

I posted on this subject recently, but there was a misunderstanding on my side. Since my French is not very good, I misread the Galois's original paper. So let me explain my question again. Let ...
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57 views

Groups of isometries

I recently read the popular scientific book "Symmetry and the Monster" and it emphasizes groups as the sets of symmetries of geometrical objects. So I was wondering, do all groups appear as symmetry ...
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285 views

How many elements are there in the group of invertible $2\times 2$ matrices over the field of seven elements?

How many elements are there in the group of invertible $2\times 2$ matrices over the field of seven elements? Sorry I have no idea so nothing to say? Any clue!
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25 views

Cardinality of $m\Bbb Z_n = \{\overline {ma} : a \in \Bbb Z_n\}$

Let $m,n \in \Bbb Z^+$ such that m divides n. I'm trying to find the cardinality of $m\Bbb Z_n = \{\overline {ma} : a \in \Bbb Z_n\}$. So, I think #$(m\Bbb Z_n)= \frac n m = k$. I tried to prove by it ...
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Uniqueness of the direct product decomposition of inclusions of finite groups

This post is a generalization of Uniqueness of the direct product decomposition of finite groups. Here we look inclusions of finite groups $(H \subset G)$ instead of just finite groups. Definition: ...
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65 views

What is the number of subgroups of order $7$?

$G$ be a simple group of order $168$. What is the number of subgroups of order $7$?
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57 views

homomorphism $\phi$ : $\mathbb Z$ $\rightarrow$ $\mathbb Z$ using $\phi(n)$=$2n$

A question from Visual group theory says : consider the homomorphism $\phi$ : $\mathbb Z$ $\rightarrow$ $\mathbb Z$ by $\phi(n)$=$2n$,where the group operation on $\mathbb Z$ is '+'. Would $\phi$ be ...
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Symmetries on sets of strings

My question is a reference request about symmetries on sets of strings. I'm not a mathematician, so the terminology I use below is probably very non-standard. My apologies. Terminology. Let $[n] = ...
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35 views

Why $|G|$ even implies $|A(G)|$ also even?

Let $G$ be finite group with even order. Why has the set $A(G)=\{g\in G: g\neq g^{-1}\}$ an even number of elements?
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17 views

Why don't semi-direct products determine a group uniquely?

While reading some group theory notes I came up to this fact: Proposition: If $G$ is the inner semi direct product of $H,K$ ($G=HK$, $H\cap K=\left\{1\right\}$ and $H\unrhd G$) then $G\cong ...
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78 views

Prove that $a^n \cdot a^m = a^{n+m}$

Let $a$ be an element of a group $G$. Prove that $a^n \cdot a^m = a^{n+m}$ for any integers $m,n \in \Bbb Z$.
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48 views

Does there exists a homomorphism for any groups $G$ and $H$

This is a question from Exercise 8.2 of Visual Group Theory which says:determine whether true or false. For any group $H$ and $G$,there is some homomorphism from $H$ to $G$. For any groups $H$ and ...
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89 views

A group of order $120$ has a subgroup of index $3$ or $5$ (or both)

What I have tried that number of $2$-sylow subgroup can be $1,3,5$ or $15$.I have solved the problem when the number of $2$-sylow subgroup is $1,3,5$. But I am not able to solve it for $15$. Any help ...
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174 views

Isomorphism of $\mathbb{Z}/n \mathbb{Z}$ and $\mathbb{Z}_n$

Let $n\in \mathbb{Z}^+$. How do I prove that $\mathbb{Z}/n\mathbb{Z}$ is isomorphic to $\mathbb{Z_n}$? Is there any good homomorphism $\phi$ I could use that graphs $\mathbb{Z}/n\mathbb{Z}$ to ...
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44 views

Examples for infinite Hamiltonian group

During teaching some basic concepts about a Hamiltonian group, I was asked about an infinite sample. According to what D.J. Robinson cited, we have a very good ...
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65 views

Map from real numbers to real numbers as abelian groups

What is an example of an abelian group homomorphism from $\mathbb{R}$ to $\mathbb{R}$, respecting addition, that is NOT a linear transformation? I'm trying to understand to what extent the scaling ...
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42 views

Is $\cos \frac{5\pi}{8}+i\sin\frac{5\pi}{8}$ in $U_8$, the multiplicative group of the $8$th roots of unity in $\mathbb{C}$?

I encountered this problem while reading Fraleigh, A First Course in Abstract Algebra, 4/e. (p.60 #19) Let $U_8$ be the multiplicative group of the $8$th roots of unity in $\mathbb{C}$. The question ...
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Is every finite group of isometries a subgroup of a finite reflection group?

Is every finite group of isometries in $d$-dimensional Euclidean space a subgroup of a finite group generated by reflections? By "reflection" I mean reflection in a hyperplane: the isometry fixing a ...
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523 views

Can $S^2$ be turned into a topological group?

I know that $S^1$ and $S^3$ can be turned into topological groups by considering complex multiplication and quaternion multiplication respectively, but I don't know how to prove or disprove that $S^2$ ...
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Software or tool for investigating groups

I'm interested in software that has the ability to investigate finite groups. In particular, I'd like to be able to ask it questions like "What are the solutions to $x^3 = 1$?" (i.e. find cube roots ...
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32 views

Left Cosets of Cyclic Subgroup

Question from a GRE Math book that I'm having trouble understanding: Find the number of left cosets of the cyclic subgroup generated by (1, 1) of $$Z_{2} \times Z_{4}$$ where Zn denotes the cyclic ...
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Amalgams of two nontrivial group is trivial

I got this question in the book Tree by Serre. Let A=$Z$. $G_1=PSL(2,Q)$ and $G_2=Z/2Z$. We take $f_1: A\rightarrow G_1$ to be an injective (I do not know what is this injective map ?) and ...
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Homology group of 3-fold sum of projective planes

I want to calculate the homology group of the 3-fold sum of projective planes defined by the labelling scheme $aabbcc$. For this I will use the following corollary from Munkres: Corollary 75.2: Let ...
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For an associative binary operation with identity, the set of invertible elements forms a group

Let $S$ be a set, and $*$ an associative binary operation on $S$. Suppose there is an element $e\in S$ such that ($1$) $e*x=x$ and $x*e=x$ for all $x\in S$. (a) Prove that there is a unique element ...
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Calculate quotients of $\mathbb S_3$ and $\mathbb D_4$

Problem Calculate all the quotients by normal subgroups of $\mathbb S_3$ and $\mathbb D_4$,i.e., charactertize all the groups that can be obtained as quotients of the mentioned groups. For the case ...
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271 views

Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions?

I'm reading a paper about Hamilton's discovery of quaternions and it explains why he failed in his 'theory of triplets' where he tried to make a vector with $3$ dimensions, as an analogy to the ...
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31 views

Homology group of space $X$ given by the labelling scheme $aabcb^{-1}c^{-1}$

I have to calculate the homology group of the quotient space $X$ given by the labelling scheme $aabcb^{-1}c^{-1}$ and then determine to which of the following spaces it is homeomorphic: $S^2, ...
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21 views

External semidirect product application

I am trying to find all normal subgroups of $\mathbb D_n$. I've read here Normal subgroups of dihedral groups that one could show the external semidirect product $(\mathbb Z/n\mathbb Z) \rtimes ...
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left regular representation of a group thru group action

Let G be a group and let $g\cdot:G\to G$(i.e.$g\cdot g'=gg'$) . This induces a permutation representation of the group. I was trying to walk thru the problems in dummite and foote. One of the problem ...
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Alternative way of proving the subgroup of rotations is normal in $\mathbb D_4$

I've just solved a basic group theory exercise which is: decide if $\{1,r,r^2,r^3\}$ is a normal subgroup of $\mathbb D_4$ (I mean the dihedral group of $8$ elements, not the one of $4$). I've used ...
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$G=\{f_n(x):n\in \mathbb{Z}\}$ is cyclic

Define $f_n(x)=x+n \;\;\forall n \in \mathbb{Z}$. Let $G=\{f_n:n\in \mathbb{Z}\}$. I proved that $G$ is a subgroup of $S_\mathbb{R}$ ($f_n$ is a permutation of $\mathbb{R}$), and now I am trying to ...
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Proving certain groups are normal

Given the following subgroup of $\mathbb S_4$: $$K=\{\mathrm{id},(12)(34),(13)(24),(14)(23)\}$$ prove that $K \unlhd \mathbb A_4$, $K \unlhd \mathbb S_4$ I am trying to solve this problem doing as ...
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When do two configurations of points belong to the same Euler Equivalence Class?

When can I say, of two or more configurations of points in a plane, that they belong to the same Euler Equivalence Class? From Euler's rotation theorem, I gather that two configurations of points are ...
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Number of non isomorphic groups of order $122$, My attempt through Sylow theory.

$|G|=122 = 2 . 61$ No. of sylow $2$ subgroups $= 1$ or $61 = n_2$ No. of sylow $61$ subgroups $= 1 = n_{61}$ Let the group of order $61$ be $H_{61}$ and the group of order $2$ be $H_2$ Then : ...
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What can you say about the order of this element?

I am self-studying algebra and encountered the following problem: If $b^{-1}ab=a^2$ and $c^{-1}ac=a^3$ and $b,c$ has orders $4$ and $3$ respectively, what can you say about the order of $a$? In ...
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Explain rings and is [S, /, -] a ring?

Okay, so we are going to use the base set of numbers [i], which contains all possible cases of ai, where a is any real number. Here are 4 possible groups on this set --> [i,*]... [i,+]... [i,/] ...
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In a finite cyclic group of order n, number of solutions to $x^m = e$ - Fraleigh p. 68 6.53,54

(53.) Show that in a finite cyclic group G of order n, written multiplicatively, the equation $x^m = e$ has exactly m solutions $x$ in G for each $m \in \mathbb{N}$ that divides n. (54.) With ...
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Checking understanding on proving uniqueness of identity and inverse elements of a group.

Sorry for such a trivial question, but just wanted to check my understanding. When proving a statement, for example, that the inverse of a group element is unique (in elementary group theory) one ...
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Let $f$ and $g$ be two group homomorphisms from $G$ to $G'$. Is $H$ a subgroup of $G$?

Can someone please verify this? Let $f$ and $g$ be two group homomorphisms from $G$ to $G'$. Let $H \subset G$ be the subset $\{x \in G: f(x)=g(x)\}$. Is $H$ a subgroup of $G$? Let $e$ and $e'$ ...
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homomorphism between diedral group $D_3$ triangle isometries and $S_3$ identification problem

My question deals with the dihedral group $D_3$ of equilateral triangle 123 (1 top vertex, 2 bottom right vertex, 3 bottom left vertex). R1 is the counterclockwise rotation of 120 degrees. R2 is the ...
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Series of multi sets

Given that a set of numbers $K = \{n_1, n_2, n_3, ... \}$. Multiple subsets are formed by randomly extracted numbers from $K$. Then series are formed by extracting numbers from the subset orderly. ...
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Cardinality of $\text{Aut}(G\times G) $

Let $G$ be a finite group. If $|\text{Aut}(G)|$ is known, what can we say about $|\text{Aut}(G\times G)|$ ?
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Solvable implies quotient group is solvable: Proof check.

I'd like to check the veracity of my proof. I've seen several proofs using different methods (some I'm allowed to use with lots of element-pushing and others using ideas I'm not allowed), but none ...
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Coprime Commutators in locally finite groups

Let $G$ be a locally finite group. Let $X = \{[a,b] \mid (\mid a \mid, \mid b \mid) =1, a,b \in G\},$ where $(m,n)$ denotes the greatest common divisor of the elments $m$ and $n$. Let $K = ...
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No proper subgroup of finite index [duplicate]

Show that $(\mathbb{Q},+)$, the group of rational numbers under addition, has no proper subgroup of finite index. Can someone please provide a proof!
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How to generate a point cloud with known symmetry?

So I would like to know if there are any published algorithms to generate point clouds with known symmetry groups, such as $D_{3h}$ or $O_h$ and stuff like that. I know lots and lots of point clouds ...
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Conditions for a group to be isomorphic to semidirect product of its subgroups

Let $G$ a group and $N$ a normal subgroup of $G$. If $G$ it have a subgroup $H$ s.t. $H \cap N $ is the trivial subgroup and $H$ is isomorphic to $G/ N$ then $G$ is isomorphic to $N\rtimes H$. Could ...