A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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condition for short exact sequence of groups to be isomorphic

Let $G$ be a group and $K_1,K_2$ be two distinct normal subgroups of $G$. We have two short exact sequences: $$1 \to K_1 {\rightarrow} G {\rightarrow} G/K_1 \to 1$$ $$1 \to K_2 {\rightarrow} G ...
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lattice-ordered group

I am just starting to read the lattice-ordered group. I was trying to prove one of the elementary properties. Give an $\ell$-group $G$, for all $a, b$ and $c$ in $G$, $a + \textit{sup} (b, c) = ...
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3answers
871 views

Groups - order of elements (element of order 3)

I need to prove there is no group that contain only one element of order 3. im don't know where to start because I don't really know anything about the group. will appreciate your help
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297 views

Properties of Group $\to$ Monoid $\to$ Semigroup.

In How to get a group from a semigroup Arturo Madigin writes: So if we look at the composition of the forgetful functors $\bf Group \to \bf Monoid \to \bf Semigroup$, we obtain a right adjoint by ...
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2answers
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For $Q$ the quaternion group, is $Q/Z(Q)$ a group? For which operation?

I'm new to group theory and am having trouble grasping the concept of cosets. For $N < G$, what exactly is $G/N$? As an example consider the quaternion group $Q$ and it's center $Z = \{1, -1\}$. ...
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1answer
85 views

Exact sequence from $G=G_0\supset G_1\supset G_2\supset\cdots\supset G_r=\{e\}$

This is Exercise 5.3 from Algebra: Chapter 0. Given a normal series of subgroups \begin{equation}G=G_0\supset G_1\supset G_2\supset\cdots\supset G_r=\{e\}, \end{equation} construct an exact ...
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134 views

Coset representatives that form a group?

$G$ is a subgroup of the Euclidean group which translational subgroup $N\triangleleft G$ (the isometries that represent pure translations) is a lattice in Euclidean space. If $H$ is a subgroup of $G$ ...
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169 views

In a group of Möbius transformations, does discontinuity imply discreteness?

Let $G$ be a subgroup of the group of Möbius transformations $$ z \mapsto \frac{az+b}{cz+d}.$$ What is the relationship between the two conditions: (1) $G$ being discrete. (2) $G$ acting properly ...
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211 views

A subgroup such that every left coset is contained in a right coset.

Let $G$ be any group, and $H \leq G$ a subgroup. Suppose that for each $x \in G$, there exists a $y \in G$ such that $xH \subseteq Hy$. In other words, every left coset of $H$ is contained inside ...
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98 views

Chief series of a group

I need help in checking some reasoning in an answer. Let $G$ be a group with order 180. Suppose G is a group with chief series $G = G_0 \geq G_1 \geq G_2 \geq \cdots \geq G_r = \{1\}.$ What are the ...
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How many ways can one paint the edges of a Petersen graph black or white?

How many ways can one paint the edges of a Petersen graph black or white? I know that the symmetrygroup of the Petersen graph is $ [S5][1]$. Furthermore this this seems like a case where I should ...
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1answer
163 views

on finite 2-group of nilpotency class two

Let $G$ be a finite 2-group such that $\mid Inn(G)\mid=4$, $Z(G)$ is not cyclic and $Z(G)$ has at least an element of order 4. Then prove or disprove there exists an $\alpha\in Aut(G)$ such that ...
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1answer
137 views

Semidirect products of groups (and its relation to the coproduct)

Suppose I have two groups $G,H$ and a homomorphism $H\rightarrow\text{Aut}(N)$. Suppose further that I have another group $T$ and two maps: $f_G : G\rightarrow\text{Aut}(T)$ and $f_H : ...
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1answer
88 views

How does one create a rotation about a given axis in $R^{3}$ from rotations about the other axes?

I was told that you can rotate a vector about a given axis in Cartesian space by combining rotations about the other two axes. I found a quick method for 90 degree rotations but I'm unsure how to ...
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1k views

Problem from Herstein on group theory

The problem is: If $ G $ is a finite group with order not divisible by $ 3 $, and $ (ab)^{3} = a^{3} b^{3} $ for all $ a,b \in G $, then show that $ G $ is abelian. I have been trying this for a ...
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3answers
204 views

Check in detail that $\langle x,y \mid xyx^{-1}y^{-1} \rangle$ is a presentation for $\mathbb{Z} \times \mathbb{Z}$.

"Check in detail that $\langle x,y \mid xyx^{-1}y^{-1} \rangle$ is a presentation for $\mathbb{Z} \times \mathbb{Z}$." Exercise 27.5 from "Groups and Symmetry" M.A.Armstrong. This should be an easy ...
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1answer
100 views

Properties of $H\lhd N(H)$ for $H\subset S_n$

Suppose we are given a subgroup $H$ of $G=S_n$. What are some techniques for studying the structure of $N_{G}(H)/H$, especially in the case that $H=\langle \sigma,\tau\rangle$? Note: $\sigma,\tau$ ...
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1answer
82 views

Computations with Normalizers [closed]

Suppose we are given a group $G$ whose structure is well-understood, and suppose we are given a small subgroup $H$. What are some techniques for studying the structure of $N_{G}(H)/H$? Sorry for the ...
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4answers
791 views

What is an additive group?

Is an additive group a group which only has an addition operation, or can it also have other operations on it? Thanks
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89 views

Group of distance preserving transformations of the plane is isomorphic to $\mathbb{Z} \ltimes \mathbb{Z}$

"Let $G$ be the group of distance preserving transformations of $\mathbb{R^2}$ which is generated by $(x,y)\mapsto(x+1,y)$and $(x,y)\mapsto(-x,y+1)$. Prove that $G$ is isomorphic to the semidirect ...
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198 views

Model theory in group theory

I am interested in useful results for group theorists that can be shown using model theory. For example : Theorem: Let $\langle X \mid R \rangle$ be presentation of a group $G$ with $X$ finite and ...
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2answers
193 views

Exponent of a direct product of cyclic groups

I have an answer to a homework question that I am not sure is correct. The question is show that if $G \cong C_{n_1} \times C_{n_2} \times \cdots \times C_{n_k}$ for positive integers $n_1, n_2, ...
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432 views

Normal subgroups of the group of invertible $2×2$ upper triangular matrices

Let $G$ be the group of all invertible $2×2$ upper triangular matrices (under matrix multiplication). Pick out the normal subgroups of $G$ from the following: (a) $H=\{A\in G:a_{12}=0\}$; (b) ...
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57 views

identification of $GL_n(\mathbb{R})/H$

could any one just tell me can I identify the group $GL_n(\mathbb{R})/H$ with the group $(\mathbb{R}^{+},.)$? where $H=$ Normal subgroup of matrices with positive determinant. .Any correct answers ...
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541 views

Which families of groups have interesting formulas for the number of elements of given order?

Suppose that $G$ is a group and that $n$ is a positive integer diving the order of $G$. Let $f_n(G)$ be the number of elements satisfying $x^n = 1$ in $G$. According to a theorem of Frobenius, then we ...
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2answers
167 views

A question about groups and subgroups.

I am working from these lecture notes. For this example, Example. List the elements of the cyclic subgroup of $\mathbb{Z}_8\times \mathbb{Z}_{15}$ generated by $(6,10)$. $$ ...
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1answer
734 views

How to find a normal abelian subgroup in a solvable group? [duplicate]

Possible Duplicate: A Nontrivial Subgroup of a Solvable Group If $H$ is nontrivial normal subgroup of the solvable group $G$, then how can I show that there is a nontrivial subgroup $A\leq ...
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2answers
220 views

A problem about group of order $p(p+1)$.

Let $G$ be a group of order $p(p+1)$ where $p$ is an odd prime and $n_{p}(G) = |\text{Syl}_{p}(G)| > 1$. The problem is to count the number of elements of $G$ that do not have order $p$. The ...
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A few questions about nonabelian cohomology of finite groups.

I apologize in advance if these questions are broad or basic. I tried to read about them at the Wikipedia, but everything is written in the language of category theory, in which I have had no formal ...
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max-n is not inherited by normal subgroups

This is exercise 3.1.9 at page 70 of Robinson, A course in the theory of groups. Prove that the property max-n is not inherited by normal subgroups, proceeding thus: let A be the additive group of ...
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2answers
82 views

On a special normal subgroup of a group

Let $G$ be a group such $H$ is a normal subgroup of $G$ and $Z(H)=1$ and $Inn(H)=Aut(H)$. Then prove there exists a normal subgroup $K$ of $G$ such that $G=H\times K$.
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Can someone explain the precise difference between of direct sum and direct product of groups?

As far as I know, the direct product of groups $G_1, \dots , G_n$ is the group with the underlying set being the cartesian product and the operation done component wise. It's not clear to me what a ...
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1answer
137 views

How could I describe $\mbox{Hom}_{\mbox{Grp}} \left({\mathbb{Z}}/{n \mathbb{Z}},{\mathbb{Z}}/{m \mathbb{Z}} \right)$?

I am trying to do this homework : Describe $$\mbox{Hom}_{\mbox{Grp}} \left({\mathbb{Z}}/{n \mathbb{Z}}, {\mathbb{Z}}/{m \mathbb{Z}} \right).$$ My road(s) so far : I note $\pi_n : \mathbb{Z} \to ...
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Examples of familiar, easy-to-visualize manifolds that admit Lie group structures

I have a trouble learning Lie groups --- I have no canonical example to imagine while thinking of a Lie group. When I imagine a manifold it is usually some kind of a $2$D blanket or a circle/curve or ...
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143 views

On automorphisms group of order $p^n$

Let $G$ be a finite group, such that $\mid Aut(G)\mid=p^n$. Then prove $G$ is p-group or $G\cong P\times C_{2}$, where $P$ is a p-group. Thank you
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Symmetries of a graph

Determine the number of symmetries in the following graph: What are the general things you should do when finding such symmetries? Usually I would label all of them from $1...n$ and note their ...
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63 views

Schroeder-Bernstein Theorem for groups [duplicate]

Possible Duplicate: When can a pair of groups be embedded in each other? Let $G,H$ be two groups. Let $f:G\rightarrow H$, $g:H\rightarrow G$ be injective group homomorphisms. From the ...
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2answers
426 views

Non-Abelian simple group of order $120$

Let's assume that there exists simple non-Abelian group $G$ of order $120$. How can I show that $G$ is isomorphic to some subgroup of $A_6$?
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1answer
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Aut($G$) $\simeq$ Aut($M$) $ \times$ Aut($N$)

A question in group theory: Let $ G = M \times N $ be the direct product of $ 2 $ normal subgroups. If $( | M | , | N | ) = 1 $ then Aut($G$) $\simeq$ Aut($M$) $ \times$ Aut($N$). I proved that ...
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1answer
863 views

Order of an element in the factor group divides order or element

Let $N$ be a normal subgroup of a finite group $G$, and $a \in G$ is an element of order $o(a)$. Prove that the order $m$ of $aN$ in $G/N$ is a divisor of $o(a)$. Here what I did: ...
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1answer
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Direct product of group order 2

If $P$ is group of order 2, how many subgroups (trivial and proper) has the group $P \times P \times P$? Labelling the elements of $P$ to be $e$ and $a$, list the proper subgroups.
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Group theory with direct product

If $(G,*)$ and $(H,.)$ are groups, we can form a new group which is called the direct product $G \times H$ of $G$ and $H$, where the combination of two elements is defined by ...
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A finite abelian group G has a subgroup of order d for all d dividing the order of G

Use Cauchy’s Theorem and induction to prove that a finite abelian group $G$ has a subgroup of order $d$ for all $d$ dividing the order of $G$.
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256 views

The importance of Schur-Zassenhaus theorem

I've just studied the Schur-Zassenhaus theorem (here there is the statement), but I don't understand its importance in the theory of finite groups. Wikipedia for example says: The Schur–Zassenhaus ...
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190 views

Groups and subgroups

if $G$ is a finite group. $H,K$ are subgroups of $G$ and $|H|=18$, $|K|=25$ than why the intersection of $H$ and $K$ is only the unit element and can't include more elements?
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Involutions and Abelian Groups, II.

In the thread Involutions and Abelian Groups, I supplied a solution to the following interesting problem (with the help of hints provided by the OP). Let $ G $ be a finite group and $ I(G) $ the ...
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3answers
356 views

$G$ is a non-abelian group with nontrivial centre $C$. Then the centre of $G/C$ trivial?

$G$ is a non-abelian group with nontrivial centre $C$. Then the centre of $G/C$ trivial? I have no idea please help me.
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105 views

How to prove that a group is simple only from its class equation

Can someone give me a hint/solution how to prove that a group is simple if its class equation is $60=1+15+20+12+12$ ?
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Idealclassgroup for quadratic field

I have got a question about an ideal class group, namely the group of $\Bbb{Q}(\sqrt{-185})$. I can say the following: I can give a representant system of the group I can name the class number: ...
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60 views

Questions around the number of subgroups of a $p$-group

Let $G$ pe a $p$ group. I have to show that the number of nonnormal subgroups is divisible by $p$ the number of subgroups differs from the number of normal subgroups by a power of $p$. ...