0
votes
0answers
14 views

Cyclic Factor group abelian proof [duplicate]

Show that if G is nonabelian, then the factor group G/Z(G) is not cyclic. I started to prove this via contrapositive. If G/Z(G) is cyclic, then G is abelian. I'm messing around with elements and ...
0
votes
1answer
33 views

Cyclic and abelian groups

Just looking for the criteria which I would use to say if these groups are cyclic. Like a short proof? for (i), (ii), (iii), (iv) (v) Thank you.
4
votes
1answer
135 views

Abelian group generators and relations

(a) Define what it means for an abelian group to be finitely generated. Explain the terms elementary divisors and rank of $G$ and describe the structure theorem for finitely generated abelian ...
0
votes
0answers
18 views

Fourier Analysis and applications to Abels groups

Find all functions $f:A\rightarrow C$ such that: $$\sum_{x\in A}|(f*f)(x)|^2 = |A|(\sum_{x\in A}|f(x)|^2)^2$$
3
votes
2answers
80 views

Give an example of a non-abelian group G containing a proper normal subgroup N such that $G/N$ is abelian.

Give an example of a non-abelian group G containing a proper normal subgroup N such that $G/N$ is abelian. I KNOW THERE IS A QUESTION OF THE SAME NAME. However, I need more involved assistance. My ...
0
votes
1answer
202 views

Question about Finite Abelian Groups [duplicate]

Let $(G, .)$ be a finite abelian group, $G=\{x_1, ..., x_n\}$ and let $x=x_1. \cdots. x_n$. Show that $x^2=e$ Suppose $G$ has no element of order 2 or that $G$ has more than one element of order 2. ...
0
votes
3answers
118 views

Commutator Subgroup is Normal Subgroup of Kernel of Homomorphism

Please help to understand this problem. Let $G$ be a group, $H$ an abelian group, $\phi : G \rightarrow H$ a homomorphism. Show that $C(G) \lhd \mathrm{Ker}(\phi)$ I must be misunderstanding ...
1
vote
1answer
87 views

Non-abelian group in which $\forall_{a,b\in G} (ab)^3=a^3b^3$ [duplicate]

Give an example of a non-abelian group, in which $(ab)^3=a^3b^3$ for every element $a,b$ in $G$. I understand that such a group should be of order divisible by 3 (see Problem from Herstein on group ...
3
votes
3answers
97 views

Isomorphisms of direct products of finite abelian groups

Suppose $G_1, G_2, H_1, H_2$ are finite abelian groups with $G_1 \times G_2 \cong H_1 \times H_2$, and $G_1 \cong H_1$. Prove that $G_2 \cong H_2$. Since the groups are finite, the isomorphisms ...
1
vote
2answers
93 views

Basic Abstract Algebra - Subgroups of Abelian Group

I'm trying to prove the following: Let $G$ be an abelian group of order 72. Show that $G$ has exactly one subgroup of order 8. I know by theorem that $G$ must have at least one subgroup of order ...
3
votes
1answer
73 views

If $f\in\hbox{Hom}_{\mathbb{Z}}(\prod_{i=1}^{\infty }\mathbb{Z},\mathbb{Z})$ and $f\mid_{\bigoplus_{i=1}^{\infty } \mathbb{Z}}=0$ then $f=0$.

Prove that if $f\in \hbox{Hom}_{\mathbb{Z}}(\prod_{i=1}^{\infty }\mathbb{Z},\mathbb{Z})$ and $f\mid_{\bigoplus_{i=1}^{\infty } \mathbb{Z}}=0$ then $f=0$. I took an element of $\prod_{i=1}^{\infty ...
2
votes
1answer
117 views

Isomorphic finite abelian groups

Let $G$ and $H$ be finite abelian groups. Show that if for any natural number $n$ the groups $G$ and $H$ have the same number of elements of order $n$, then $G$ and $H$ are isomorphic. I know, ...
2
votes
3answers
657 views

Prove that a finite abelian group is simple if and only if its order is prime.

So I'm having trouble with this problem. I know that the definition of a simple group means that the group has no nontrivial subgroups. I know that this can be proven somehow with the help of the ...
2
votes
3answers
99 views

Does $\Bbb Q/ \Bbb Z$ have a proper subgroup that is not finite?

Does $\Bbb Q/ \Bbb Z$ have a proper subgroup that is not finite? I suspect it does not. However since we could take a subgroup of all $p$ sets $\{\frac{1}{p} + \Bbb Z\}$ if we consider $p$ to be ...
-1
votes
2answers
103 views
1
vote
1answer
185 views

If $Y: G\to H$ is a group homomorphism and $G$ is abelian, prove that $Y(G)$ is also abelian.

What I got: Suppose $Y$ is a homomorphism and that $G$ is abelian. Then for all $a,b \in G$, $ab=ba$, and thus $Y(ab)=Y(ba)=Y(a)Y(b)=Y(b)Y(a)$. However, this seems too simple and I was confused on ...
0
votes
1answer
184 views

Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.

it seems like it... should be? In that I can't think of any counterexamples off the top of my head. I was looking up these http://en.wikipedia.org/wiki/Hamiltonian_group and saw the quaternion group, ...
0
votes
3answers
152 views

Prove or disprove that there is an abelian, noncyclic group of order 52.

So I've heard one must invoke Sylow's theorems in order to break down something like this. So far I know that there is a subgroup of order 13 in G, and that it's the only subgroup of order 13 in G. To ...
2
votes
3answers
279 views

Prove that $H$ is a subgroup of an abelian group $G$

Let $H = \{x \in G: x = y^2$ for some $y\in G\}$; that is, let $H$ be the set of all the elements of $G$ which have a square root. Prove that $H$ is a subgroup of $G$, where $G$ is an abelian group. ...
4
votes
1answer
256 views

Express quotient of free abelian group as direct sum of cyclic groups

This is the problem: Let $G$ be the quotient of the free abelian group with $\mathbb{Z}$-basis $x_1, x_2, x_3$ by the subgroup $H = \langle x_1 + 3x_2, x_1 + 4x_2 + x_3, 2x_1 + 5x_2 + x_3\rangle$. ...
4
votes
2answers
580 views

Order of products of elements in a finite Abelian group

We want to show that if $a,b\in G$ where $G$ is a finite Abelian group, we have $\operatorname{LCM}(|a|,|b|) = |ab|$ given that $ab \neq e$. How I approached this question was by saying let ...
3
votes
1answer
504 views

A group $G$ is Abelian iff $(ab)^n = a^n b^n$ for all $a,b \in G$ and $n \in \Bbb Z$

Prove that $G$ is Abelian if and only if $(ab)^n =a^n b^n $ for all $a, b \in G$ and $ n \in \mathbb{Z} $. I used proof by induction in the $ \rightarrow$ direction of the proof and I'm done with ...
12
votes
2answers
233 views

$G$ abelian group with $\vert G\vert =mn$ and $\gcd(m,n)=1$.

I hope this is not a duplicate. First of all, in what follows I'm not allowed (unfortunately) to use the structure theorem for abelian groups. I'm asked to prove the following: Let $G$ be an ...
0
votes
1answer
181 views

Classify the abelian groups of order 81, 144 and 216

Classify the abelian groups of order n, 2n, 4n
4
votes
2answers
260 views

If the order of a finite abelian group is not divisible by a square, show that the group must be cyclic.

If the order of a finite abelian group is square free, show that the group is cyclic. This is a question from "basic abstract algebra" by bhattacharya
4
votes
3answers
108 views

Help finding all elements of order 2 in $S_6$.

I am trying to find all elements of order 2 in $S_6$. I am trying to understand how to achieve this. Here is my attempt. We need only count the number of permutation of the forms $ (a_1 a_2)\\ ...
-4
votes
2answers
145 views

Homorphism Being Trival when Group Is Abelian [on hold]

Let $G$ be a group, and let $g \in G$. The function $\gamma_g\colon G \to G$ defined by $(\forall a \in G)\colon \gamma_g(a)=g ag ^{-1} $ is an automorphism of $G$. The automorphisms $\gamma_g$ are ...
7
votes
2answers
362 views

Abelian group admitting a surjective homomorphism onto an infinite cyclic group

I am working on the following problem: Let $G$ an Abelian group and $f: G \to \Bbb Z$ a surjective homomorphism. Prove that $G \cong \ker(f) \times \Bbb Z$ By means of the First Isomorphism ...
1
vote
2answers
85 views

Calculate $\operatorname{Hom}_{\Bbb Z}(\Bbb Z_6,\Bbb R^*\oplus \Bbb C^*)$ [duplicate]

Let $\Bbb R^*=\Bbb R-\{0\}$ (non-zero real numbers) and $\Bbb C^*=\Bbb C-\{0\}$ (non-zero complex numbers) be multiplicative groups. Is this equality true? $$\operatorname{Hom}_{\Bbb Z}(\Bbb ...
10
votes
2answers
169 views

The existence of a group automorphism with some properties implies commutativity.

Let $G $ be a finite group, $T$ be an automorphisom of $ G $ st $ Tx = x \iff x=e $. Suppose further that $ T^2 =I $. Prove that $ G $ is abelian. I was thinking if I show $ T aba^{-1} b^ ...
0
votes
1answer
52 views

In direct products of $n$ groups, do we also prove conditions for a group to be abelian or not? [duplicate]

If we let $G_1,...,G_n$ be groups, When proving that the direct product $G_1 \times .... \times G_n$ is abelian if and only if each of $G_1,...,G_n$ is abelian, can someone please help me Im ...
6
votes
4answers
1k views

Show that $({\mathbb{Q}},+)$ is not finitely generated using the Fundamental Theorem of Finitely Generated Abelian Groups.

Can anyone please help me out on how to use the fundamental theorem of finitely generated abelian groups to prove the above question?
3
votes
3answers
77 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 ...
4
votes
2answers
242 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 ...
10
votes
2answers
811 views

Find an abelian infinite group such that every proper subgroup is finite

I found this question in Arhangel'skii and Tkachenko's book Topological Groups and Related Structures. The first chapter of the book is devoted to algebraic preliminaries. The question actually ...
1
vote
1answer
108 views

Prove abelian non-cyclic p-group of order p^m (m>2) has a non-cyclic proper subgroup without structure theorem

It should have a copy of $\Bbb Z/p\Bbb Z\times \Bbb Z/p\Bbb Z$ but my brother's class has not learned it yet and this is for a homework problem... Any help?
1
vote
1answer
564 views

In an abelian Group in which there are 2 subgroups of order m and n there is a subgroup of order mcm(m,n), how to prove?

This is an Herstein's exercise If an abelian group has two subgroups of order m and n, prove that it also has a subgroup of order lcm(m,n). I've solved the exercise right before this, which is ...
2
votes
1answer
216 views

p-group: cyclic $n \leq 1$ | abelian $n \leq 2$

$p$ prime number, $n$ a non-negative integer, $G$ group. (a) $\forall G$ with $|G| = p^{n}$ cyclic $\Leftrightarrow$ $n \leq 1$. (b) $\forall G$ with $|G| = p^{n}$ abelian $\Leftrightarrow$ $n \leq ...
5
votes
2answers
211 views

Subgroups of $\Bbb{R}^n$ that are closed and discrete

I am trying to prove that every closed discrete subgroup of $\Bbb{R}^n$ under addition is a free abelian group of finite rank. I have tried to do this by induction on the dimension $n$. Base ...
3
votes
1answer
104 views

CEP for Abelian groups and lattices

An algebra $A$ has the congruence extension property (CEP) if for every $B\le A$ and $\theta \in \operatorname{Con} B$ there is a $\phi \in \operatorname{Con} A$ such that $\theta = \phi \cap (B\times ...
3
votes
1answer
362 views

Torsion-free quotient group of an abelian group

Let $G$ be an abelian group, and let $H\leq G$. Prove that if $G/H$ is torsion free, then $H$ contains the torsion group of $G$. Proof: Let $x\neq1$ be an element in the torsion group. Thus there ...
3
votes
3answers
239 views

Check if $(\mathbb Z_7, \odot)$ is an abelian group, issue in finding inverse element

Take $\mathbb Z_7$ and the operation $\odot$ defined on it as follows $\forall a,b \in \mathbb Z_7$: $$\begin{aligned} a \odot b=a+b+3\end{aligned}$$ Check if $(\mathbb Z_7, \odot)$ is a group and ...
5
votes
1answer
299 views

Freiman homomorphism on generating set

I got stuck in an exercise from Tao and Vu's book Additive Combinatorics. It is ex. 5.3.4. on page 226. In the following let (Z,+) and (W,+) be two abelian groups and let A $\subset$ Z and B ...
0
votes
1answer
100 views

Finitely generated abelian group has a regular normal form

Prove that every finitely generated abelian group admits a regular normal form. I am having some trouble getting my head wrapped around this problem. If anyone can offer suggestions or help it would ...
7
votes
3answers
813 views

Finite abelian $p$-group with only one subgroup size $p$ is cyclic

My goal is to prove this: If $G$ is a finite abelian $p$-group with a unique subgroup of size $p$, then $G$ is cyclic. I tried to prove this by induction on $n$, where $|G| = p^n$ but was not ...
3
votes
2answers
145 views

Simple properties of a direct product

I am working on some homework for modern algebra class. The problem I just finished seems relatively easy, but I have learned to be wary of that feeling when it comes to this material. Below are the ...
6
votes
2answers
900 views

If $H$ is a normal subgroup of $G$ and if both $H$ and $G/H$ are abelian, is $G$ abelian?

Pretty straightforward: If $H$ is a normal subgroup of $G$ and if both $H$ and $G/H$ are abelian, is $G$ abelian?
1
vote
0answers
203 views

Normal abelian subgroup of a solvable group [duplicate]

Possible Duplicate: A Nontrivial Subgroup of a Solvable Group How to find a normal abelian subgroup in a solvable group? Could someone help me with this proof? Let $G$ be a solvable group ...
1
vote
1answer
276 views

Torsion free abelian group of rank 1

I find it hard to understand a part of a proof on torsion free abelian groups of rank 1. Let $A$ and $B$ be torsion free groups of rank 1 and of the same type. Let $a'$ and $b'$ arbitrary non ...