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

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Basic question Subgroups

I need to find all the values of $a$ and $n$ that gives $\{0,a\}$ is a subgroup of the group $(\mathbb{Z}_n,+)$. Assum $n \geq 2$ and $a \neq 0$. Actually I thing that for each $a$ and $n$ we will ...
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171 views

Structure of the group of arithmetic functions

This question was originally posted in Elements of finite order in the group of arithmetic functions under Dirichlet convolution. and it goes as follows: Let G be the group consisting of all ...
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1answer
58 views

On the structure of maximal parobolic subgroups of orthogonal groups over finite fields

Let $q=p^f$ be an odd prime power and $P$ be a maximal parobolic subgroup of $GO^\varepsilon(n,q)$ stabilising a totally singular $k$-subspace. It is known that $P$ has shape $A{:}(B\times C)$, where ...
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380 views

Number of elements of given order in a group

Example 1: "Calculate the number of elements of order 2 in the group $C_{20} \times C_{30}$" To do this, I split the groups into their primary decompositions and got that the groups with elements of ...
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7answers
816 views

Quotient Group G/G = {identity}?

I know this is a basic question, but I'm trying to convince myself of Wikipedia's statement. "The quotient group $G / G$ is isomorphic to the trivial group." I write the definition for left ...
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1answer
175 views

Calculating the number of elements of some order of a direct product of groups

My question is: Consider a finite group $G$. For any integer $m \geq 1$ set $\gamma(m) = \gamma_G(m)$ the number of elements $g \in G$ such that ord($g$) = $m$. We say that $m$ is a possible order ...
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5answers
242 views

In which of the finite groups, the inverse of Lagrange's Theorem is not correct?

This is a multiple choice for finite groups. For which one of the following groups, the converse of Lagrange's Theorem is not generally satisfied? I know the converse is true for cyclic groups. 1) ...
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116 views

An example of a group

I need an example of a finite group $G$ by the following properties: 1) Order $G$ is $336$. 2) For every prime $p$, $G$ has not any elements of $7p$. 3) the number of Sylow $7$-subgroups $G$ is ...
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614 views

when do two elements of SO3 commute?

I have an intuition that at least one of the element should be in the form of diagonal matices with diagonal entries being 1 or -1 (e.g I3) I don't know if there's any other possiblity Please give ...
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56 views

Proving that every element in a monoid occurs once [duplicate]

Possible Duplicate: Proving that every element of a monoid occurs exactly once let (B,*) defines a monoid with a finite number of elements Let the elements of B be x1,x2,x3,x4 where every ...
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100 views

Equivalence of $x,y\in G$ given that $xzy^{-1}z^{-1}$ is a commutator for some $z$

Let $G = \langle a,b,c\:|\: a^2, b^2, c^2\rangle$. Let $\tilde{}$ by the equivalence relation on $G$ generated by conjugation and inversion (i.e., $x\tilde{} y$ if there is a finite sequence of ...
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201 views

Calculating the index of a subgroup in a free group

Can someone help me with the following question? Let $F=\langle x_1,\ldots,x_n \rangle $ be a finitely-generated free group . Let $p$ be a fixed prime, and $H$ be the normal subgroup of $F$ ...
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4answers
478 views

Proving the stabilizer is a subgroup of the group to prove the Orbit-Stabiliser theorem

I have to prove the OS theorem. The OS theorem states that for some group $G$, acting on some set $X$, we get $$ |G| = |\mathrm{Orb}(x)| \cdot |G_x| $$ To prove this, I said that this can be written ...
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2answers
202 views

Group of order $24$ containing no elements of order $6$

Let $G$ be a group of order $24$ with no elements of order $6$. Let $T$ be a subgroup of $G$, is a Sylow $3$-subgroup. I have prove that $G$ has no normal subgroup of order 2, so it is also clear that ...
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94 views

abelian finite groups - basic

G is the set of all subset of A. (For example - Say $A=\{1,2,3\}$ than $G=(\{1\},\{2\},\{3\},\{1,2\}...)$. ($A$ is at east two different elements). the binary operation $*$ is intersection. I need to ...
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89 views

Why $U\le A \times B$ does not imply $U=\left(A\cap U\right) \times \left( B \cap U \right)$?

Let $A,B$ be groups.Can you explain why $U\le A \times B$ does not imply $U=\left(A\cap U\right) \times \left( B \cap U \right)$ this is an exercise in the book of the theory of finite groups an ...
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259 views

How can I show the free abelian group of rank $r$ is isomorphic to an $r$-copy of $\mathbb{Z}_\infty$?

Today, this problem was given to me. Let $F$ be a abelian free group of rank $r$. Show that it is isomorphic to an $r$-copy of $Z_{\infty}$. I could do some messy job about it but so far I ...
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1answer
68 views

Finite group with given order

Is there any finite group $G$ with order $p(p^2-1)$ and normal subgroup $N$ such that $G/N\cong PSL( 2, p)$?
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98 views

New subgroup has twice the size of the old one?

While going through my class notes, I came across a statement I copied down from the board that I don't quite understand. The statement is this: Let $H$ be a subgroup of a group $G$. Suppose that $g ...
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88 views

Proving that every element of a monoid occurs exactly once

let $(B,\star)$ defines a monoid with a finite number of elements Let. the elements of $B$ be $\{x_1,x_2,x_3,x_4,\cdots\}$ where every element of $B$ occurs exactly once in this list let $y$ be the ...
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154 views

Very generic question about Commutator and Center

Let $G$ be a finite group and $G'$ the commutator group of $G$. What can I say about $G' \cap Z(G)$? Could you be as specific as possible about p-Groups?
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206 views

Structure of the centralizer of an element in Sym(n)

Do we know the structure of the centralizer of any element in $S_n$? Or the centralizer of a permutation which has $q$ disjoint $r$-cycles where $r\leq n$. If we know, can anyone give proof or ...
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962 views

G acts on G/H by left muliplication - Orbit, stabilizer and fixed points?

Reference: p. 8 http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/gpaction.pdf This PDF doesn't unfold all the steps hence can someone please notify me of bungles? Thank you. I tried: $G$ ...
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3answers
265 views

group multiplication table

I really looked all over the web and searched for an example I will understand. I don't understand how to complete a multiplication table! (all examples I found the Identity element was given) ...
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81 views

abelian and finite group

G= $Q^+$ (Rational numbers diffrent from zero) $a*b = ab/2$ I already proved this is a group now I need to prove or disprove that it is abelian and or finite group. For abelian - from what I ...
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4answers
104 views

Is the bijection shown by commutativity?

"Let $G_1. G_2$ be groups. Prove that the map $$ \varphi: G_1 \times G_2 \rightarrow G_2 \times G_1$$ $$ \varphi: (g_1, g_2) \mapsto (g_2, g_1)$$ defines an isomorphism between $G_1 \times G_2$ and ...
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5answers
173 views

Is $\{-2,2\}$ a group under $a\star b=\max\{a,b\}$?

Lets say $G={-2,2}$ and $a*b=\text{max}\{a,b\}$. I need to check if this is a group and if it does than is it abelian or not and finite or not. Well... first, I'm not sure if this is a group. for ...
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2answers
246 views

Elements of finite order in the group of arithmetic functions under Dirichlet convolution.

Let $(G, ∗)$ be the group of arithmetic functions $f : N \to C$ that satisfy $f (1)\neq 0$, with group operation given by the Dirichlet product $∗$. The identity function $I$ is the identity element ...
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233 views

Every simple group of odd order is isomorphic to $\mathbb{Z}_{p} $ iff every group of odd order is solvable

I'm trying to prove the following claims are equivalent: Every simple group of odd order is of the type $\mathbb{Z}_{p}$ for prime $p$ Every group of odd order is solvable. Getting from 2 to 1 was ...
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127 views

Groups with 3 conjugacy classes

How to find (describe) all groups which have 3 conjugacy classes? Thanks in advance!
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60 views

Let $N$ be a normal subgroup of group $G$ and $G=(N\times C_{3})\rtimes C_{2}$. Then prove $G=N\times (C_{3}\rtimes C_{2})$.

Let $N$ be normal subgroup of $G$ and $G=(N\times C_{3})\rtimes C_{2}$. Then prove $G=N\times (C_{3}\rtimes C_{2})$. Thank you
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Permutation Group as quotient of Free Group

I could use some help with the following question: Let $S_{n}$ be the permutation group of $\left\{ 1,...,n\right\}$ , what is the minimal $k\in\mathbb{N}$ such that $S_{n}$ is a quotient of ...
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371 views

Faithfulness - Group Action on Left Cosets by Left Multiplication

p. 6: http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/gpaction.pdf Pretend the blue set was not given and I have to calculate it myself: For all $x \in G, f(x) = xgH$ is faithful $\iff ...
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3answers
135 views

Are these factor groups cyclic?

а) $\mathbb{Q}/2\mathbb{Z}$ b) $\mathbb{R}^*/\mathbb{Q}^*$ How to find out (and prove) if the factor groups above are cyclic or not? Thanks in advance!
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160 views

Dehn presentation proof reference request

Can someone give me a reference for a proof that the Dehn presentation of a knot group gives us the fundamental group of the knot complement in $S^{3}$?
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212 views

Group Action - Permutation on the Polynomial

I'm trying to check the permutation on the polynomial is a Group Action, but I'm not getting the second axiom. I'm following my lecturer's work --- Examples 2.1 and 2.6 on page 5 on ...
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2answers
311 views

Nonabelian $p$-groups all of whose proper subgroups are abelian.

Theorem. Let $G$ be a finite, non-abelian $p$-group all of whose proper subgroups are abelian. Then $|G'|=p$. Take a counterexample of minimal order. Assume that exist a $H$ such that ...
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68 views

Linear algebra about cyclic group and torsion factors

Express the commutative group $Z^{3}/(f_{1}.f_{2},f_{3})$ as a direct sum of cyclic group where $f_{1}=(4,6,9), f_{2}=(2,4,12), f_{3}=(4,8,16)$ my answer is $Z[x]/(-11x^{2}+22x-128)$. I wonder if my ...
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2answers
120 views

1 dimensional representations of $S_n$

I want to show that $S_n$ has only two 1 dimensional represnetations. mainly the trivial and sign represnetations. Where I assumed that our Field we're working on is with characteristic $\neq 2$. ...
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1answer
65 views

Conditions for a $\mathrm{Hom}$ group to be finite.

If $G$ is a finite group, and $D$ is a divisible abelian group, what are some conditions on $D$ for which $\mathrm{Hom}(G,D)$ is finite? At first I thought that having $D$ with finite torsion ...
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72 views

Question about commutator

Let $A$ be an abelian normal subgroup of $G$ and $x\in G$. How can we prove the following? (a) The mapping $A\mapsto A$ given by $a \mapsto [a,x]$ is a homomorphism. (b) $[A,\langle ...
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638 views

Surjective group homomorphism

If we have two group presentations, can we say that we have a surjective group homomorphism $f: G \rightarrow H $ if the generators of $G$ map to all the generators of $H$ and all the relations of $G$ ...
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Prime divisor in the Automorphism group

Suppose that $G$ is a non-abelian simple group and $r$ is a prime number such that $r$ not divide $|G|$. It is a question for me that whether $r$ can divide |Aut($G$)|?
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Subgroup of a soluble group is soluble

I'm trying to show that if $G$ is a soluble group with $H$ some subgroup then $H$ is also soluble. My argument is as follows: As $G$ is soluble then we have the subnormal series: $\{e\}\triangleleft ...
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132 views

Examples of non-isomorphic abelian groups which are part of exact sequences

Suppose $A_1$, $A_2$, $A_3$ and $B_1$, $B_2$, and $B_3$ are two short exact sequences of abelian groups. I am looking for two such short sequences where $A_1$ and $B_1$ is isomorphic and $A_2$ and ...
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When are $((C_2 \times C_2) \rtimes C_3) \rtimes C_2$ and $((C_2 \times C_2) \rtimes C_2) \rtimes C_3$ isomorphic?

Let's consider $\mathfrak{G}:=((C_2 \times C_2) \rtimes_{\phi} C_3) \rtimes_{\nu} C_2$ (which I do believe is $\mathcal{S}_4$, please confirm or argue against) and $G:=((C_2 \times C_2) \rtimes_{\mu} ...
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how can I find the qutient of this subgroup?

Let $G$ be a finite group and $N\lhd G$. For every $x\in G-N$ we have $$C_G(x)N/N\leq C_{G/N}(Nx)$$ where $C_G(x)$ is the centralizer of $x$ in $G$. But I don't know what is the qutient. In fact I ...
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906 views

Centralizer, Normalizer and Stabilizer - intuition

What is the motivation/intuition behind these concepts? What notion/property of a group do they capture? Or what is the scenario of application. Thanks.
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Constructions of the smallest nonabelian group of odd order

I write $|X|$ for the number of elements in a finite set $X$. Recall some basic facts: If $p$ is a prime number, then any group $G$ of order $p^2$ is abelian. Sketch of proof: Fix a prime $p$ ...
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128 views

Bound on the number of p-groups for fixed exponent

It's well-known that for each prime number $p$ there are exactly two groups of order $p^2$, five of order $p^3$, and fifteen of order $p^4$ (at least when $p>3$). I know that the classification of ...