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

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left and right coset verifications

How to find the left and right cosets of the subgroup $H = \{r_0, s_0\}$ of $D_4$? And are they the same? If we let $H' = \{r_0,r_2\}$, are the left and right cosets the same, where $H'$ is a ...
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476 views

Order of $ab$ if $a$ and $b$ commute and $\langle a\rangle\cap\langle b\rangle=\{1\}$

So, one of the homework problems I am working on is If $a$ and $b$ are elements of a group that commute and $\langle a\rangle\cap \langle b\rangle = \{1\}$, what is the order of $ab$ if the order ...
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421 views

Rank of a cohomology group, Betti numbers.

How is the rank of a cohomology group computed and what does it convey? I am trying to understand the concept behind betti numbers in a simplicial homology. Edited with details: Given a set of ...
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619 views

Follow-up to question: Aut(G) for G = Klein 4-group is isomorphic to $S_3$

This is most likely a lack of understanding of wording on my part. I was considerind the Klein 4-group as the set of four permutations: the identity permutation, and three other permutations of four ...
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259 views

Property of minimal normal subgroups 2

I'm having some problems with understanding the corollary of the next theorem : Every minimal normal subgroup $K$ of $G$ is a direct product $K = T_1 \times T_2 \times \cdots \times T_k$ where ...
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107 views

what is the order of AB?

Let $A$ and $B$ be any invertible $4 \times 4$ matrices with $0$ and $1$ everywhere, and let $H=\{A^n| n \in \mathbf{Z}\}$, $N=\{A^nB^m | n,m\in \mathbf{Z} \}$ are the subsets $H$ and $N$ subgroups ...
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113 views

A subgroup of a direct product

Let $H$ and $K$ be subgroups of a group $G$. Suppose $H=A\times B$. Does it follow that $H\cap K=(A\cap K)\times (B\cap K)$? I'm having a hard time trying to prove that $H\cap K\le (A\cap ...
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202 views

Composition series of a group

If H is normal in a group G, where G has a composition series, then G has a composition series one of whose terms is H.
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146 views

divisible and torsion groups

I'm stuck in this problem: If $A$ is an abelian group which is torsion free, prove that there exists a divisible group $D$ and an injective homomorphism $A \to D$. Same question also but for $A$ ...
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271 views

Embedding of symmetric group in general linear group

Recently, one question was posed on embeddings of finite groups in $GL(n,\mathbb{Z})$ (here); it is mentioned that $S_n$ can be embedded in $GL(n-1,\mathbb{Z})$. How to prove it? Can we replace ...
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99 views

The permutation group corresponding to translations in three direction on a discrete lattice

What is the name of the permutation group corresponding to all the translation operations in the $3$ directions $x$, $y$ and $z$ (with periodic boundary conditions) of a general rectangular discrete ...
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293 views

Finite subgroups of orthogonal transformations in $\mathbb{R}^3$

The group of orthogonal transformations of $\mathbb{R}^3$ is direct product of the group of rotations and the group $Z=\langle x\mapsto -x\rangle\cong \mathbb{Z}/2$. The finite subgroups of group of ...
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94 views

Clarifications on proof that the fixed points of order $p$, $i_p(G)\equiv -1\pmod{p}$

I'm reading this paper by Marcel Herzog on jstor: http://www.jstor.org/stable/2040939?seq=1 I want to follow up on a few things about the short proof of Theorem 1, found on the bottom of page 1 of ...
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191 views

induced action on quotient space

Let $X$ be a topological space on which a group $G$ acts . let $N$ and $K$ be subgroups of $G$. under what condition we have an induced action of $K$ on $X/N$? My guess: if $N$ is normalized by ...
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75 views

The relationship between $G_i H / G_{i+1} H$ and $G_i /G_{i+1}$

Suppose that $G$ is a group, $G_{i+1} \triangleleft G_i \triangleleft G$, $H \triangleleft G$. I need to investigate the relationship between the groups $G_i H / G_{i+1} H$ and $G_i /G_{i+1}$, but so ...
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231 views

Functor from a group to a poset

Are there examples of functors from the category of a single group to the category a partially ordered set (some sort of representation of the group in a poset) ?
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232 views

conjugate subgroups of the symmetric group

Let $S_n$ be the symmetric group. let $H=S_{d_1}\times\cdots \times S_{d_k}$ such that $d_1+\cdots + d_k=n$. What does the following statement means: every two embeddings of $H$ in $S_n$ are ...
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335 views

conjugate subgroups to a given group

let $G$ be a group. Is it true that the only subgroup of $G$ that is conjugate to $G$ is $G$ itself? if $G$ is finite this is clear as conjugate subgroups have the same order but what about infinite ...
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200 views

The cycle structure of the permutation $a \mapsto ma \bmod{n}$

Given an odd $n$, and an $m$ such that $(n,m)=1$, i would like to know what is the cycle structure of the permutation $\pi_{n,m} (a)=ma\bmod{n}$. Specifically, how do i know if $\pi_{n,m}$ and ...
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2k views

What is the order of the element in group theory?

I know the order of the group is the number of elements in the set. For example the group of $U_{10}$ (units of congruence class of 20) has order 4. Major Edit, kinda changed the question. Lets say ...
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83 views

“noncommutativity” of arranging 12 distinct objects in (4 bunches of 3) or (3 bunches of 4)

Suppose you have a set of twelve distinct objects, and you were interested in ways of arranging them into a 3 by 4 rectangle. You wonder: does it matter if I arrange them in 3 bunches of 4 or 4 ...
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420 views

Conjugates Generating a normal subgroup

This problem is taken from I.N. Herstein Problem If $G$ is a group and $a \in G$ if of finite order and has only a finite number of conjugates in $G$, prove that these conjugates of $a$ generate a ...
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43 views

Guidance and sanity check needed - question on the isomorphism theorems

The question is from Joseph .J Rotman's book - Introduction to the Theory of Groups and it goes like this: $A,B,C$ are subgroups of $G$, so $A\leq B$, prove that if $(AC=BC\ \text{and}\ A\cap ...
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30 views

If $\lvert g \rvert=m$ is finite then prove that $ng=0$ if and only if $m\mid n$.

Let $G$ be an abelian group and let $g \in G$. If $\lvert g \rvert=m$ is finite then prove that, for $n\in \mathbb Z$, $ng=0$ if and only if $m\mid n$. I think this amounts to proving that: $$ng=0 ...
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32 views

To finde the center of $D_4$

is there a nice/smart way to find the center of $D_4$? rather then going through every element?
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25 views

Do isomorphisms preserve simplicity?

This is a very simple* question that I surprisingly can't find the answer to, and I am too stupid to come up with a counterexample or a proof. So does simplicity of one group and an isomorphism ...
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33 views

Infinite Suzuki Groups

Often I found myself on a symbol like $Sz(F)$ where $F$ is an infinite field. What is the definition of an infinite Suzuki group? Are they linear groups? Where I could find some informations about ...
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40 views

A more swift method for Conjugation Classes

I am asked to find the conjugation classes of a group order n. I am aware what a conjugation class is and how to find it. My question: is there a quicker/more simple way to find the conjugation ...
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34 views

What is the center of the Valentiner group $\mathcal{V}=\langle I, Q \rangle$?

(Please refer to this question first: Is $\langle(26543),(34)(56),(12)(3654)\rangle $ isomorphic to $A_6$? ) I want to understand the center of the Valentiner group: $$\mathcal{V}=\langle I, Q ...
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Internal and Direct Product question, need help with explanation

I am asked to express a group G={1,7,17,23,49,55,65,71} under multiplication modulo 96 as an external and internal direct product of cyclic groups. However I also have an example to help with it, but ...
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33 views

Transitive Actions

Given that transitive actions are in a bijection with conjugacy classes of subgroups of G, describe isomorphism classes of transitive actions for the following groups: $C_4, Z/8, C_2 × C_2, S_3$ Can ...
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35 views

Homomorphism preserves normality

let $\phi:G\rightarrow G'$ be a homomorphism, and let N' be a normal subgroup of G'. I want to show that $\phi^{-1}[N']$ is also normal subgroup of G. My work : since homomorphism preserves ...
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Show that a certain normal subgroup or a product is abelian

Let $A$ be a normal subgroup of $G\times H$. If the identity of $G$ is $1_G$ and the identity of $H$ is $1_H$, $(x,y)\in A$ has the property that $x\ne1_G$ and $y\ne1_H$ unless $(x,y)=(1_G,1_H)=1_A$. ...
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47 views

If G is finite p-group and H is a subgroup, show that there is a composition series that contains H.

Let $p$ be a prime number, suppose $G$ is a finite p-group and let $H$ be a subgroup of $G$. Show that there is a composition series that contains $H$. I have already shown that if $G$ if a finite ...
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38 views

greatest common divisor proof

I have two questions about a prove that I have to do for my mathematic study. I'm now thinking about it the whole day, but can't find the prove. Let $a,b \in \mathbb Z_{>0}$. (a) Prove: $\gcd(2^a ...
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23 views

Dimension of subspace stabilized by group and principle character

Let $\phi: G \rightarrow GL(V)$ be a representation with character $\chi$. Let $W$ be the subspace $\{v \in V : \phi(g).v = v$ for all $g \in G \}$ of $V$. Prove that $\dim W = \langle \chi, ...
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22 views

Finding free subgroups thanks to Lie algebras

Let $f : F \to G$ be a homomorphism from a free group $F$ to a group $G$. I heard that, in order to verify whether or not $f$ is one-to-one, it is possible to associate a Lie algebra $E_0^*(H)$ to any ...
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35 views

Show that $a^m$ is in $H$ for every $a$ in $G$ [duplicate]

Let $H$ be a normal subgroup of $G$, and let $m=(G:H)$. Show that $a^m \in H$ for every $a \in G$. I have been thinking about this question for a few days but I get something informal. What am ...
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21 views

CHECK: Show that $N_G(aHa^{-1})=aN_G(H)a^{-1}$ for $a \in G$.

Show that $N_G(aHa^{-1})=aN_G(H)a^{-1}$ for $a \in G$. Hint: When working this problem, I found that in showing the two sets equal I had to be extremely careful. Try not to make big jumps. At one ...
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31 views

Show that a polynomial f(x) over a field k is irreducible if and only if the polynomial f(x + 1) is irreducible.

I am very much unsure what definitions and formulas are relevant for this question. I've toyed around with the lemma "An element a ∈ R is a root of a polynomial f ∈ R[x] if and only if (x − a) divides ...
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27 views

Prove that for each $\sigma \in Aut(S_n)$ $n \neq 6$. $\sigma(1,2)=(a,b_2),\sigma(1,3)=(a,b_3), …, \sigma(1,n)=(a,b_n)$

I am reposting it getting insufficient help from the previous post (Although I got some hint) Prove that for each $\sigma \in Aut(S_n)$ $n \neq 6$. $\sigma(1,2)=(a,b_2),\sigma(1,3)=(a,b_3), ...., ...
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33 views

Show a bijection between two inverse images of a homomorphism

$ \Phi: G \rightarrow H $ is a group homomorphism. There are $ h,h' \in H $, so that $ \Phi^{-1}(\{h\})$ and $\Phi^{-1}(\{h'\})$ are not empty. Show a bijection between $ \Phi^{-1}(\{h\}) $ and $ ...
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How do we compute |G| = |Ox| · |Gx|?

I was given a set X and a group G and was asked to find Gx and Xg. Then I was asked to find the G-equivalence class of X for each of the G-sets which is the orbit Ox but i'm having trouble verifying ...
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44 views

Understanding Group Action

In general are we just supposed to make an educated guess about what the $G$-set is for a group action is if it's not specified? Here are two examples of what I mean. I am asked to find a fixed ...
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33 views

Eigenspace of finite abelian group

Let $\rho: G\to {\rm GL}_n(\mathbb{C})$ be faithfull representation of finite abelian group $G$ and $V$ is the eigenspace of some $g\in G$. Is it true that $V$ is also eigenspace for all $G$ (that ...
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34 views

How to check which subgroups of $D_4$ are normal

How do I check which subgroups of $D_4$ are normal? Trying all elements seems very cumbersome. So far, I know only basic theorems like Lagrange's and the homomorphism as well as the isomorphism ...
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40 views

Prove that two finite abelian groups having the same number of elements of order n are isomorphic. [closed]

Let $G$ and $H$ be finite abelian groups. Prove the following assertion: If for each $n \in \mathbb N$ the groups $G$ and $H$ have the same number of elements of order $n$, then $G\cong H$. I ...
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28 views

Find all the cosets of Dihderal group 6 with subgroup H

Let H={$\rho_{0}, \rho_{2}, \rho_{4}$}, a subgroup of D6, the group of symmetries. Where $\rho_{0}$=identity permutation, $\rho_{2}$=(1,3,5)(2,4,6) and $\rho_{4}$=(1,5,3)(2,6,4). How would you ...
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28 views

Verification of Proof that if G is not abelian G/Z(G) is not cyclic

I will prove this by the contrapositive: "If G/Z(G) is cyclic then G is abelian" Proof: We assume that G/Z(G) is cyclic. This means it is generated by a left coset $(aZ(G))^n$=e for some integer n. ...
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34 views

Question on abstract algebra about Group?

I need an explanation, why $ (\mathbb{Z}_7,\oplus _6 )$ is not a Group? As I have discovered so far. The following conditions are satisfied I) Closed! II) Associative! III) ...