Articles of abelian groups

Every infinite abelian group has at least one element of infinite order?

Is the statement true? Every infinite abelian group has at least one element of infinite order. I am searching for an infinite abelian group with all elements having finite order. Please help me to find such groups.

Divisible abelian $q$-group of finite rank

What does “finite rank” mean in the context of divisible abelian $q$-group? A divisible abelian $q$-group of finite rank is always a Prüfer $q$-group or it can be also a finite product of Prüfer $q$-group? Thanks for all future answers to my questions.

Classifying $\mathbb{Z}_{12} \times \mathbb{Z}_3 \times \mathbb{Z}_6/\langle(8,2,4)\rangle$

I wish to classify $\mathbb{Z}_{12} \times \mathbb{Z}_3 \times \mathbb{Z}_6/\langle(8,2,4)\rangle$ according to the fundamental theorem of finitely generated abelian groups. We have that it is of order $72$. Based on previous help received here, I have attempted to set it up as a matrix: $$\begin{bmatrix}12 & 3 & 6\\8&2&4\end{bmatrix}$$ But I am unable to find any […]

Proof of elliptic curves being an abelian group

What are some simple proofs that the points on an elliptic curve form an abelian group under addition? I am mostly looking for proofs of closure and associativity, since the other three requirements follow immediately from the definitions. Note that I am specifically looking for simpler proofs, since I am only still in high-school, so […]

How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?

This question already has an answer here: Determining the number of subgroups of $\Bbb Z_{14} \oplus \Bbb Z_{6}$ 1 answer

Betti numbers and Fundamental theorem of finitely generated abelian groups

My textbook (author : fraleigh) says that Fundamental theorem of finitely generated abelian groups Every finitely generated abelian group $G$ is isomorphic to a direct product of cyclic groups of the form $$\Bbb{Z_{(p_1)^{r_1}}}\times \Bbb{Z_{(p_2)^{r_2}}}\times\dots\times \Bbb{Z_{(p_n)^{r_n}}}\times\underbrace{\Bbb{Z}\times\Bbb{Z}\dots\times\Bbb{Z}}_{\text{r times, r : betti number}}$$ where $p_i$ are primes, not necessarily distinct, and $r_i$ are positive integers. The prime powers […]

The order of a non-abelian group is $pq$ such that $p<q$. Show that $p\mid q-1$ (without Sylow's theorem)

This question already has an answer here: Groups of order $pq$ without using Sylow theorems 5 answers

The following groups are the same.

This is an exercise from J.J.Rotman’s book: Prove that the following groups are all isomorphic: $$G_1=\frac{\mathbb R}{\mathbb Z},G_2=\prod_p{\mathbb Z(p^{\infty})}, G_3=\mathbb R\oplus\big(\frac{\mathbb Q}{\mathbb Z}\big)$$ What I have done is: Since $tG_1=\frac{\mathbb Q}{\mathbb Z}$, which $t$ means the torsion subgroup; and the fact that $G_1\cong tG_1\oplus\frac{G_1}{tG_1}$ so I should show that $\frac{\mathbb R}{\mathbb Q}\cong\mathbb R$. A theorem […]

$H_1 ,H_2 \unlhd \, G$ with $H_1 \cap H_2 = \{1_G\} $. Prove every two elements in $H_1, H_2$ commute

This is the proof, which I mostly understand except for one bit: You have $h_1 \in H_1$ and $h_2 \in H_2$. We also have $h_1^{-1}(h_2^{-1}h_1h_2) \in H_1$, because $h_2^{-1}h_1h_2 \in h_2^{-1}H_1h_2 = H_1$. Similarly, we have $(h_1^{-1}h_2^{-1}h_1)h_2 \in H_2$. Therefore $$ h_1^{-1}h_2^{-1}h_1h_2 \in H_1 \cap H_2 = \{1_G\} $$ and so $h_1^{-1}h_2^{-1}h_1h_2 = \{1_G\}$. Let’s […]

Pontryagin dual of the unit circle

Define the unit circle as $\frac{\mathbb{R}}{2\pi\mathbb{Z}}.$ I know the Pontryagin dual (looking at properties of the Fourier transform on locally compact Abelian groups) is $\mathbb{Z}$ but why? Any notes or suggestions will be appreciated.