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

learn more… | top users | synonyms (2)

0
votes
0answers
11 views

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 ...
0
votes
1answer
22 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, ...
1
vote
4answers
44 views

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 ...
0
votes
0answers
17 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 ...
1
vote
1answer
17 views

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 ...
0
votes
2answers
40 views

$G=\{f_n(x):n\in \mathbb{Z}\}$ 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 ...
0
votes
2answers
50 views

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 ...
0
votes
0answers
15 views

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 ...
2
votes
1answer
24 views

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 ...
0
votes
2answers
45 views

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 : ...
2
votes
1answer
60 views

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 ...
2
votes
1answer
21 views

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 ...
6
votes
1answer
257 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 ...
1
vote
1answer
38 views

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,/] ...
0
votes
0answers
23 views

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. ...
9
votes
1answer
109 views

Cardinality of $\text{Aut}(G\times G) $

Let $G$ be a finite group. If $|\text{Aut}(G)|$ is known, what can we say about $|\text{Aut}(G\times G)|$ ?
1
vote
1answer
25 views

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 ...
3
votes
2answers
53 views

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 ...
1
vote
0answers
20 views

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 = ...
2
votes
0answers
48 views

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 ...
1
vote
0answers
29 views

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!
1
vote
0answers
22 views

Proving $|G|=pq$ and $p>q$ , $q$ does not divide $p-1$ $\implies$ $G$ is cyclic , without using Cauchy's and Sylow's theorems [duplicate]

Without using Cauchy's or Sylow's theorems , can we give a proof of the result that "If $ p,q$ are primes such that $p>q$ and $q$ does not divide $p-1$ , then any group of order $pq$ is cyclic " ? ...
1
vote
0answers
32 views

divisible subgroup without axiom of choice

the theorem asserting that the divisible subgroup of an Abelian group is a direct summand depends on Zorn's lemma. in ZF without AC is there a construction which yields a model of an Abelian group ...
0
votes
0answers
13 views

Compounding unary operators

I am working with the symmetric group $S_5$. I have 3 unary operators defined: $R$, $T$, and $O$, and I'm writing about their composition. Suppose I want to denote the compound operation of "$T$, ...
3
votes
1answer
48 views

Something that is true for every element of $\text{Sym}(\Bbb{N})$

I'm trying to prove that: Every element of $\text{Sym}(\Bbb{N})$ can be written as $f\circ g$ for some $f,g\in \text{Sym}(\Bbb{N})$ with $f^2=g^2$. But I can't even prove this for ...
3
votes
1answer
134 views

Proof that group is commutative if every element is its inverse (feedback wanted)

This is one of my first proofs about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. $e$ denotes the identity element. ...
1
vote
1answer
56 views

every Abelian group is a converse lagrange theorem group

Let $G$ be a finite abelian group, then $G$ has a subgroup of order $n$ if and only if $n\mid G$. Proof: by Lagrange if $H\leq G$ then $|H|$ divides $|G|$ so this proves one of the implications. We ...
-4
votes
1answer
92 views

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 ...
2
votes
2answers
26 views

If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. show that $H$ is contained in every sylow $p$ subgroup of $G$

If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. show that $H$ is contained in every sylow $p$ subgroup of $G$ Attempt: $|H|=p^k \implies |G|=p^{n_1} q^{n_2} ...
3
votes
1answer
45 views

Groups - Prove that every element equals inverse of inverse of element

This is my first proof about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. We use $^{-1}$ to denote inverse ...
0
votes
1answer
38 views

Defining a group from edge set of graph

I consider three islands represented by vertices V and the travel routes by ship are represented by the edges E. Here G=(V,E). I consider the non-empty set E and define the binary operation ...
1
vote
1answer
33 views

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 ...
0
votes
2answers
36 views

Show that if $G$ is a group of order $168$ that has a normal subgroup of order $4$ , then $G$ has a normal subgroup of order $28$

Show that if $G$ is a group of order $168$ that has a normal subgroup of order $4$ , then $G$ has a normal subgroup of order $28$. Attempt: $|G|=168=2^3.3.7$ Then number of sylow $7$ subgroups in $G ...
0
votes
1answer
27 views

Proof the existence of a normal subgroup

Let $K$ be a normal subgroup of $H/N$, and $N$ be a normal subgroup of $H$. Show that there is $M \lhd H$ such that $N \subset M$ and $K=M/N$. I have some difficulties to prove it.
3
votes
1answer
25 views

Automorphism of Tree

Let $\sigma$ and $\theta$ be two automorphisms of tree $X$. I want to show that min$_{v\in V(X)}d(v,\sigma(v))=$min$_{v\in V(X)}d(\theta^{-1}\sigma\theta(v),v)$. I know every automorphism of tree is ...
2
votes
3answers
86 views

If $G$ is a simple $f$ an homomorphism, and $A\lhd H$ is such that $[H:A]=2$, show $f(G) \subset A$

I'm stuck with the following problem. Can someone help me by providing a hint? Suppose $G$ is simple and let $f$ be an homomorphism between $G \to H$. If $\#G\ne2$, $A\lhd H$, and $[H:A]=2$. Then ...
7
votes
4answers
637 views

A group with five elements is Abelian

I tried to prove the following theorem: A group with five elements is abelian. I know only the definition of a group and a subgroup but no more.(this is a problem from Topics in Algebra by IN ...
3
votes
0answers
54 views

Tarski monster groups with more than one prime

A Tarski monster group is an infinite, finitely generated group where every proper, non-trivial subgroup is cyclic of order $p$ for a fixed prime $p$. These were shown to exist for "large enough" $p$ ...
2
votes
1answer
32 views

$G_i$ s are normal subgroups , then $\bigl[G:\bigcap_{i=1}^n G_i \bigr]\Bigm|\prod_{i=1}^n[G:G_i]$?

If $G_i$ $(i=1,\dots,n$) are normal subgroups of $G$, of finite index, then is it true that $\displaystyle\bigl[G:\bigcap_{i=1}^n G_i \bigr]\Bigm|\prod_{i=1}^n[G:G_i]$?
1
vote
1answer
27 views

Group Theory $Z_2$ representations

I am trying to understand some group theory. In the notes I am following, I am told: Recall the representations of $\mathcal{Z}_2$: Trivial: $\rho_0(e) = 1$, $\rho_0(a)$ = 1 (i) $\rho_1(e) = 1$, ...
1
vote
2answers
59 views

The multiplicative group of all the $2^n$-th roots of unity

Consider the multiplicative group $G$ of all the (complex) $2^n$-th roots of unity where $n=0,1,2...$. Which of the following statements are true? Every proper subgroup of $G$ is finite, $G$ has a ...
2
votes
1answer
39 views

Generators of the group of integers exercise

Let $a,b \in \mathbb Z$. (1) Prove that $\{a,b\}$ is a system of generators of $\mathbb Z$ if and only if $(a,b)=1$, where $(a,b)$ is the greatest common divisor between $a$ and $b$. (2)Show that ...
1
vote
2answers
34 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 ...
3
votes
0answers
32 views

If a finite group G is nilpotent then G/F is abelian, where F is the Frattini subgroup of G

Let $G$ be a finite nilpotent group. Consider $F$ the Frattini subgroup of $G$, that is, intersection of all maximal subgroup of $G$. Prove that $G/F$ is abelian. What I am trying that G/F is ...
4
votes
1answer
60 views

hint with an exercise algebra

I'm stuck with the following I hope someone could help me. Let $N$ a normal subgroup of $G$. Show that if $[G:N]=4$, exists a normal subgroup $M$ of $G$ s.t. $[G:M]=2$. My idea: Since $G/N$ has ...
3
votes
2answers
62 views

An infinite torsion group with element orders sensitive to subgroups

Is there an infinite torsion group $G$ such that for every $H\le G$ there is some $n\in \Bbb N$ with $$(\forall x\in H)(x^{n}=1), ~~(\forall x\in G\setminus H)(x^{n}\ne1)$$
6
votes
2answers
140 views

Self studying higher mathematics?

I'm fairly well-versed in calculus but I would like to explore beyond calculus. I have looked into the basics of some topics in higher mathematics such as group theory and abstract algebra and they ...
2
votes
0answers
54 views

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: ...
3
votes
4answers
91 views

If $G/K\cong H/K$ must $G\cong H$?

Let K be a normal subgroup of $H$ and $G$ so that $G/K$ is isomorphic to $H/K$, must we have $G\cong H$? I can't really tell you what I have tries since I haven't really done anything worth telling. ...
1
vote
2answers
67 views

Does $G$ always have a subgroup isomorphic to $G/N$?

Let $G$ be a group and $N$ a normal subgroup of $G$. Must $G$ contain a subgroup isomorphic to $G/N$? My first guess is no, but by the fundamental theorem of abelian groups it is true for finite ...