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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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 ...
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1answer
24 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
20 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
29 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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69 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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24 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
41 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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49 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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20 views

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
31 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
59 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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2answers
44 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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39 views

Rreferences for free groups [duplicate]

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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2answers
63 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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20 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 ...
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1answer
38 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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24 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
211 views

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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21 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
60 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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22 views

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?
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101 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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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
27 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 ...
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1answer
40 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
51 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
36 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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111 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 ...
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79 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
45 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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1answer
54 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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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
83 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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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
53 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 ...
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1answer
42 views

Composition series and direct product

I have two solvable groups $G_1$ and $G_2$ and wonder whether their direct product $G_1 \times G_2$ is solvable. I can easily write subnormal series $G_1 \times G_2 \rhd G_1 \rhd \cdots$ and $G_1 ...
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138 views

Dihedral Group D14 - Conjugancy and Subgroups

Consider $D_{14}$, the dihedral group of order $14$. This is the group of symmetries of the regular 7-gon. Label the vertices of the pentagon clockwise as $1, 2, 3, 4, 5, 6, 7$. Let $x = (1 2 3 4 5 6 ...
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55 views

group without involution is 2-divisible

Let $G$ be an arbitrary torsion group without involutions. Show that $G$ is 2-divisible. I think it is enough to show $G$=$2G$ but i can't show why $2G$ can't be proper subgroups of $G$ ? Please ...
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3answers
59 views

Let $G$ be a group. If $H\leq G$ is a subgroup and $N\vartriangleleft G$, then $HN$ is a subgroup of $G$.

Let $H$ be a subgroup of $G$ and $N$ a normal subgroup of $G$. Prove that $HN=\{hn\:|\:h \in H, n \in N\}$ is a subgroup. Here what I have so far. It is not really much I understand what I need to ...
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Check: Find all the non-isomorphic Abelian groups of order $200$.

Find all the non-isomorphic Abelian groups of order $200$. Since $200=5^2 \cdot 2^3$, there are $3 \times 2=6$ such groups. The six partitions are: $25 \times 8$ $25 \times (4 \times 2)$ $25 \times ...
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1answer
27 views

On number of elements of a particular order of $G_1 \times G_2$

If $n(G,d)$ be the no. of elements of order $d$ in a group $G$ , then can we find $n(G_1 \times G_2,d)$ in terms of $n(G_1,d)$ and $n(G_2,d)$ ? Since $o(x)=o(x,e_2)$ for any $x \in G_1$ and similar ...
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101 views

Maps to all finite cyclic groups factor implies map to integers factors

Let $G$, $H$ be groups (we lose nothing here if we assume they're abelian), let $f:G\to H$ and $g:G\to \mathbb{Z}$ be homomorphisms. This last map gives us homomorphisms $g_n:G\to {\mathbb{Z}}/{n ...
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1answer
34 views

action of $GL_3$ on $P^2$

Find the action of $GL_3(K)$ on $\mathbb P_k^2 $, and compute its orbits and also the isotropy groups for all its orbits. ($K$ is an algebraically closed field) I know that $GL_3$ acts on ...
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Is a semidirect product of linear groups a linear group?

It is known that linear groups are not closed under extensions, but what if the extension splits, i.e. it is a semidirect product? Suppose that $K,R$ are subgroups of $\mathop{GL}(n,\mathbb{F})$, ...
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CHECK: Determine the groups of order $99=3^2 \cdot 11$.

Determine the groups of order $99=3^2 \cdot 11$. The trivial group of order $99$ is $\mathbb{Z}_{99}$, which is cylic. By Sylow's theorem, the number of subgroups of order $11$ must divide $99$, and ...
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1answer
38 views

Check:etermine the number of Sylow $2$-subgroups and Sylow $3$-subgroups that $G$ can have.

Let $G$ be a group of order $48$. By the $1$st Sylow theorem $G$ has a Sylow $2$-subgroup and a Sylow $3$-subgroup. Suppose none of these are normal. Determine the number of Sylow $2$-subgroups and ...
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39 views

Intersection of composition factors

For $G=HK$, and $K= K_1\cap K_2$, both normal in $G$, if $G/K_1$ and $G/K_2$ are solvable, show that $G/K$ is solvable. By the Third Isomorphism Theorem, $$\frac{G/K}{K_i/K}\cong \frac{G}{K_i}$$ ...
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67 views

Existence of a normal subgroup in G

Today on my algebra test I had such an exercise: Let $|G|=66$. Show that there is a normal subgroup in $G$ of order $3$. I am not even sure that's true. I wanted to show that $n_{3}$=1. But from ...
6
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1answer
69 views

How many Sylow $3$-subgroups can a group of order $72$ have?

How many Sylow $3$-subgroups can a group of order $72$ have? Let $G$ be a group of order $72=2^3 \cdot 3^2$. The number of Sylow $3$-subgroups $n_3$ divides 24 and has the form $n_3=3k+1$ by the ...
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3answers
63 views

Special linear group $SL(2,\mathbb Z_3)$ doesn't have subgroups of order $12$

Prove that $SL(2,\mathbb Z_3)$ doesn't have subgroups of order $12$. I am pretty lost with this problem. I've tried to think of a morphism $$f:SL(2,\mathbb Z_3) \to Aut(\mathbb {Z_3}^2)$$ $$A \to ...