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

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How can we identify the quotient group $GL(n,\mathbb{R})/SL(n,\mathbb{R})$

How can we identify the quotient group $GL(n,\mathbb{R})/SL(n,\mathbb{R})$ I am using First isomorphisam theorem,but i am not getting the solution how can we find out the right hand side set of the ...
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1answer
49 views

Computing the quotient group

Let $k$ be a homomorphism from $\mathbb Z$ to $\mathbb Z/2\mathbb Z\times\mathbb Z$ defined by $1 \mapsto (1 + 2\mathbb Z, -2)$. Then, what is the quotient group of $\mathbb Z/2\mathbb Z\times\mathbb ...
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1answer
20 views

for any subset S of finite group G and any g,h∈G, show |gSh|=|S|

Let $G$ be a finite group. For any subset $S$ of $G$, and any group elements $g,h \in G$, I want to show that $|gSh|=|S|$, i.e. that the cardinality of $S$ is preserved when multiplying on the left ...
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1answer
70 views

Could someone check my work on this exercise

I solved the following exercise, could someone please check my work? Exercise: Let $$ A = \left ( \begin{array}{cccc} 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 ...
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2answers
57 views

Number of non-isomorphic groups of order $p^2$

The number of non-isomorphic groups of order $p^2$, where $p$ is a prime number is: 1. 1 2. $p$ 3. 2 4. $p^2$ What is simplest method to find number of non-isomorphic group? I read from various ...
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0answers
31 views

Non abelian group of order 1575

Construct a non abelian group of order $1575$. I am sure I am to use semi direct product.Give me some idea/hint to start.
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3answers
70 views

$G/Z(G)$ is cyclic useful for proving groups abelian?

It's a common exercise to prove in an abstract algebra book that if $G/Z(G)$ is cyclic then $G$ must be abelian. But I've always found the exercise strange because if $G$ is abelian then $Z(G)=G$ and ...
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1answer
59 views

random walk on finite cyclic group

Suppose that I have a random walk on the finite cyclic group of order $d > 2$, where the initial probability distribution $Q$ assigns the values $p, q, r$ to $-1, 0, 1$, respectively, where $p + q ...
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2answers
37 views

Composition series of nilpotent group

I found this problem in The Theory of Groups by Marshall Hall. Let the group $G$ be of order $p^rq^s$. If $G$ has two composition series $1 \unlhd A_1 \unlhd A_2 \unlhd \cdots \unlhd A_r \unlhd ...
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1answer
50 views

Odd order matrix in $GL_n(\mathbb F_2)$ that doesn't commute with any order $2$ matrix?

Is it the case that for all $n$, there is an invertible matrix over $\mathbb F_2$ of odd order which does not commute with any matrix of order $2$ in $GL_n(\mathbb F_2)$. I think this is equivalent to ...
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38 views

Is SL(2, 3) a subgroup of SL(2, p) for p>3?

As the title says, I was wondering whether SL(2,3) is a subgroup of SL(2,p) for p>3. I know that it is for p=5 (it can be found explicitly using the quaternionic representation), and I have some ...
2
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2answers
54 views

Suppose that $G$ is a group of order $30$ and has a Sylow $5$-subgroup that is not normal.

Suppose that $G$ is a group of order $30$ and has a Sylow $5$-subgroup that is not normal. Find the number of elements of order $1$, order $2$, order $3$, and order $5$. But this scenario can't ...
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2answers
49 views

Proving Fermat's Little Theorem in general and use that to prove Euler's Generalization of Fermat's Little Theorem

Can anyone help me with this? I know there are many different ways to do this and threads explaining this question. However I can't seem to find one that uses only group/ring theory. I haven't ...
2
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0answers
20 views

Direct product of groups and isomorphism [duplicate]

Let $A, B, C$ three groups such that $A \times C \cong B \times C$. I already know that if $A, B$ and $C$ are abelian and finite, then $A \cong B$. I think this result does not hold anymore if they ...
2
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1answer
21 views

Showing that a relation on elements of a group is an equivalence relation

Let $G$ to be group and $A,B<G$. Let $x,y \in G$. we define $x \sim y$ to be $y=a\ast x \ast b$ for $a\in A$ and $b \in B$. I want to prove that $x \sim y$ is an equivalence relation. I should ...
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1answer
17 views

Automorphisms and Mappings

Consider $Aut(Z_4 ⊕Z_2)$. Any automorphism φ is determined by where we send the two generators (1, 0) and (0, 1) of $Z_4 ⊕ Z_2$. Also, in any automorphism an element must be sent to an element with ...
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1answer
19 views

Checking class isomorphisms

I have an abelian group G of order 441. I am trying to give all possible isomorphism classes. So far I have: $Z_{441},$ $Z_{49} +Z_9 $ since 9 and 49 are coprime and + means direct product ...
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0answers
14 views

What is the Jacobian for Sim(3) lie group action on 3D points ? (4d homogenous points)

I am coding up Sim(3) constraint types for a factor graph, and need to calculate the jacobian of the Sim(3) group action on 3D points. I am following the guide on http://ethaneade.com/lie.pdf ...
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22 views

Generators of Intersection of two Subgroups

Let $G$ be a group and let $A$ be the subgroup of $G$ generated by $\{a_i\}_{i\in I}$; let $B$ be the subgroup of $G$ generated by $\{b_j\}_{j\in J}$, where $I$ and $J$ are index sets. Is there a way ...
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2answers
31 views

Number of $n-1$-dimensional subspaces of $n$-dimensional space over finite field

I got a question with two parts. Let $V$ be a $n$-dimensional vector space over $\mathbb{F}_{p}$ - finite field with $p$ elements. a) How many $1$-dimensional subspaces $V$ has. b) How many ...
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0answers
46 views

Group, do we have $(A\times B)^*=A^*\times B^*$?

Let $A,B$ two group. I denote the group of unit of $A$ and $B$ by $A^* $ and $B^*$. Do we always have $$(A\times B)^*=A^*\times B^*$$ ?
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A question on normality of $\langle x \rangle Z(G)$ for non-abelian group $G$

Let $G$ be a non-abelian group , then I have noticed that for every $x \in G$ \ $Z(G)$ , $\langle x \rangle Z(G)$ is a subgroup such that $Z(G) \subset \langle x \rangle Z(G) \subset G$ ; I would ...
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1answer
17 views

Lower Central Series and Generators

Let $G$ be a group generated by $S=\{x_1,\cdots,x_n\}$ and let its lower central series be defined as $\Gamma_1=G$, $\Gamma_m=[\Gamma_{m-1},G]$ for $m\geq 2$. By definition $\Gamma_m$ is generated by ...
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1answer
33 views

Isomorphism of Quaternion group

Prove that $Q_{8} \cong H \rtimes G \Rightarrow H=\{e\}$ or $G=\{e\}$. My proof (is it correct?): We know that (it's a fact from the lecture): If $M \cong H\rtimes G$, then $H \cong K \unlhd M$ ...
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0answers
20 views

How to search the set of papers whose references contain a given preprint?

I am reading a preprint titled Combinatorial Group Theory In Homotopy Theory I by Fred Cohen (available at Cohen's web page) Now I need to find all papers whose references contain this preprint. Is ...
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45 views

For some $\mid G \mid = 12$, then $G \cong A_4$

Im having trouble understanding why $G \cong A_4$ if $\mid G \mid = 12$. Can anyone explain it a simple way. My book says that either $G$ has a normal Sylow 3-subgroup or $G \cong A_4$($G$ has a ...
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0answers
28 views

Rreferences for free groups

I have done free groups. I studied it from Rotman two semesters back. But this semester I am doing combinatorial group theory and obviously it starts with free groups. I have to revise Free groups but ...
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39 views

Isomorphism type of a finite group with respect to multiplication modulo 65

I'm the same guy revising for my group theory exam and posted a few days ago. I'm at the chapter on Finitely Generated Abelian Groups, and my prof gave this example which I don't quite understand: ...
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16 views

Is there a complete list of forbidden minors of graph of genus 1?

Could some one please help me to find the answer of the following question? For which integer g, a complete list of forbidden minor of graphs of genus g is known? Specially is the list known for the ...
1
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1answer
27 views

How do I show that $\mathbb{R}-\{0\}/T$ is isomorphic to $\mathbb{R}^+$?

Here, $T = \{-1, 1\}$ and $\mathbb{R}-\{0\}$ is the multiplicative group of all the non-zero reals, and $\mathbb{R}^+$ is the multiplicative group of positive real numbers.
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15 views

Decomposition of SU(n) anticommutator

In $SU(N)$, the special unitary group, the algebra generators $T_a$ are hermitian and traceless. The structure constants are fixed with $[T_a,T_b]=i f_{abc}T_c$. In the fundamental representation of ...
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3answers
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When does $(ab)^n = a^n b^n$ imply a group is abelian?

Suppose the identity $(ab)^n = a^n b^n$ holds in a group for some $n\in\mathbb{Z}$. For which $n$ does this necessarily imply the group is abelian? For example, when $n=-1$ or $n=2$, the group must be ...
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0answers
17 views

using Lattice Isomorphism Theorem toto find all subgroups $\left\{H | \:SL_n\left(F_7\right)\le H\le GL_n\left(F_7\right)\right\}$

I am desperately trying to use the Lattice Isomorphism Theorem to find all subgroups $\left\{H | \:SL_n\left(F_7\right)\le H\le GL_n\left(F_7\right)\right\}$, (where $F_7$ is the prime field of 7 ...
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1answer
49 views

free abelian group

let $X=\{a_i\mid i\in I\}$ be a set , then the free abelian group on X is (isomorphic to) the group defined by the generators X and the relations (in multiplicative notation) ...
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construct non-commutative group with prescribed center

I want to construct an affine noncommutative group $G$ over a field of char zero with a prescribed center $Z(G)$. How do I do this?
3
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1answer
52 views

Question about infinite product of groups.

What is following product: $$ G = \prod_{n \in \mathbb{N}} \mathbb{Z}_{p^n} $$ where $p$ is a prime and $\mathbb{Z}_{p^n}$ is usual cyclic group of order $p^n$. Obviously it is not isomorphic to the ...
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0answers
35 views

Proof that a particular subgroup is proper

I've been stuck on this for a long time ... I'm reading a textbook which simply states "this subgroup is proper" but it doesn't make sense to me. Context: I have a pro-$p$ group $G$, which just means ...
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1answer
25 views

$|G|=2p$, $p \geq 3$ prime then $G$ is abelian or $G \cong D_{2p}$

I am doing the following problem: Let $G$ be a group such that $|G|=2p$ with $p \geq 3$ prime, then $G$ is abelian or $G \cong D_{2p}$. Suppose $G$ is not abelian. By Cauchy theorem, there exist ...
0
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1answer
37 views

Assume G is a group, x,y is in G; x and y are not identity, but $x^3=1$ and $y^2=1$ and $(xy)^2=1$. Find the order of G and the group table

So I am stuck with this problem and I can't seem to find the relationship with the x, y and identity in dealing with size of group and how they connect with $(xy)^2=1$. Can someone help me with this? ...
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2answers
46 views

Is a homomorphisim one-to-one or onto?

The definition of a homomorphism $f$ from $G$ to $H$, given by Pinter, says that: If $G$ and $H$ are groups, a homomorphism from $G$ to $H$ is a function $f: G \rightarrow H$ such that for any two ...
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1answer
28 views

Commutator of $D_{2n}$

I am trying to calculate the commutator of the Dihedral group. If $n=1,2$ then $[D_n,D_n]=1$. Now I consider the case $n\geq 3$. I thought of using the property $[G,G] \subset H$ iff $H \lhd G, G/H$ ...
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99 views

Using the Third Isomorphism Theorem

Here's my question Using the Third Isomorphism Theorem, show that if m, n are positive integers then there is an isomorphism: $\Bbb Z_m \cong \Bbb Z_{mn}/\Bbb Z_{n}$ I began this by ...
3
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1answer
65 views

Group generated by two polynomials

The $Homeo(\mathbb{R})$ be the group of all the homeomorphisms on $\mathbb{R}$, with the group operation of composition. Let $f(x)=x^3+\alpha$ and $g(x)=x^3+\beta$ be two elements of ...
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1answer
37 views

Proving a function is a monomorphism

Suppose that H is a group and there is a group homomorphism $f : D_{14} \to H$ with the property that $f(o) \not= eH$ and $f(r) \not= eH$. Show that $f$ is a monomorphism. Where $o$ is rotation and ...
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77 views

group action, cauthy theorem , orbits [closed]

group action please urgent help please? Suppose that $G$ is a group of order 25 and that $X=\{(g_1, g_2, g_3, g_4, g_5)\; : \; g_i \in G \text{ for } i=1,\dots,5 \text{ and } g_1g_2g_3g_4g_5=e \}$ ...
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1answer
35 views

Subgroup Lattice of D14 - normal and centre

I saw this question earlier on the forum and was wondering if my result to it was correct! If D14 is the dihedral group acting on a heptagon, are the only subgroups in the lattice D14, < r> , ...
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0answers
15 views

Proving equality of finitely generated subgroups of $U(3, \mathbb{Q}_2)$

Let $N = U(3, \mathbb{Q}_2)$ where $\mathbb{Q}_2$ is the ring of rational numbers of the form $m2^n$ with $m, n \in \mathbb{Z}$. Let $t$ be the diagonal matrix with diagonal entries, $1, 2, 1$ and put ...
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4answers
65 views

Proving Map is Well-defined

Okay so I have this question: Let $G$ be a group and suppose that $N$ and $K$ are normal subgroups of $G$, where $N \leq K$. Define a map: $\theta:G/N \rightarrow G/K$ by $\theta(aN)=aK$. ...
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2answers
47 views

Unique morphism from the additive group $\mathbb Q$ to $\mathbb Z$

I am trying to prove that the only group homormorphism from $\mathbb Q$ to $\mathbb Z$ is the trivial one but I couldn't Suppose there is $x \in \mathbb Q$ : $f(x)=z \neq 0$. We can write ...
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1answer
48 views

Minimal non-countable Groups

I was thinking the following thing: Is there an uncountable group whose all proper subgroups are countable which is also for instance locally soluble? I've found some example of minimal ...