Define a group homomorphism $\phi: \frac{\mathbb{Z}}{p^{r_1}\mathbb{Z}} \oplus … \frac{\mathbb{Z}}{p^{r_m} \mathbb{Z}} \to \frac{\mathbb{Z}}{p^{r_1}\mathbb{Z}} \oplus … \frac{\mathbb{Z}}{p^{r_m} \mathbb{Z}}$ by seinding each element to its multiple by $p$( $p$ is a prime number and $r_1 \geq … \geq r_m$ ). What is the image $p(\frac{\mathbb{Z}}{p^{r_1}\mathbb{Z}} \oplus … \frac{\mathbb{Z}}{p^{r_m} \mathbb{Z}})$ of such homomorphism? How can we characterize this group( […]

How would one go about proving that $\mathbb{Z}_m\oplus \mathbb{Z}_n \cong \mathbb{Z}_d\oplus \mathbb{Z}_l $ as groups, where $l=$lcm($m,n$) and $d=$gcd($m,n$)? I am attempting to use the fundamental theorem of finitely generated abelian groups but am struggling. In the interest of honesty, this is a past exam question that I am attempting for which the solutions are […]

I’ve come across another exact sequence, where (I guess) I need to deduce the result using some properties of torsion. I am calculating the homology of the Klein bottle using attaching maps. I start by defining $\Phi:I \times I \to K$ as the natural map and denote $\partial(I \times I)$ as the boundary, then let […]

Let $G$ be a finite abelian group, and let $2G$ denote the subgroup $\{ g * g : g \in G\}$. Let $G[2]$ be the 2-torsion subgroup of $G$. I want to show that $$ G/2G \cong G[2]. \qquad (1) $$ The closest I could get was to prove that $G/G[2] \cong 2G$ using the […]

This question is a follow-up of this earlier question I asked. Let $$ A_1\twoheadrightarrow A_2\twoheadrightarrow A_3\twoheadrightarrow A_4\twoheadrightarrow \cdots $$ be an inductive sequence of abelian groups, the connecting homomorphisms of which are surjective and split, that is, we have embeddings $A_{n+1}\rightarrowtail A_n$ such that the diagram \begin{array}{ccccccccc} A_n & \twoheadrightarrow & A_{n+1}\\ \uparrow & & […]

If $G$ is a finite Abelian group and for any prime $p$ divides $|G|$ there exists exactly one subgroup of order $p$ in $G$. Suppose $G_p=\{x\in G|x \text{ is a p-element}\}$, then prove $G_p$ is cyclic for every $p\text{ dividing } |G|$. The book gives a hint that because $G$ is Abelian, it is a […]

I was trying to prove that a normal, abelian subgroup of $G$, $N$ is in the center of $G$ given that $|\operatorname{Aut}(N)|$ and $|G/N|$ are relatively prime. The official question: Let $N$ be an abelian normal subgroup of a finite group $G$. Assume that the orders $|G/N|$ and $|\operatorname{Aut}(N)|$ are relatively prime. Prove that $N$ […]

Let $G\cong \Bbb{Z}_{p^{r_1}}\oplus\Bbb{Z}_{p^{r_2}}\oplus\cdots \oplus \Bbb{Z}_{p^{r_s}}$ be a finite abelian $p$-group, where $r_1\geq r_2\geq \cdots \geq r_s\geq 1$. Let $H\cong \Bbb{Z}_{p^{t_1}}\oplus\Bbb{Z}_{p^{t_2}}\oplus\cdots \oplus \Bbb{Z}_{p^{t_u}}$ be a subgroup of $G$, where $t_1\geq t_2\geq \cdots \geq t_u\geq 1$. Prove that $s\geq u$ and $r_i\geq t_i$ for each $i=1, 2, …, u$. It seems obviously. But I just can’t prove […]

Suppose $G$ is a non-abelian group and $H,K$ are two abelian subgroups of $G$. Then must $HK$ be an abelian subgroup of $G$? I know an example, but I am confused. Thus I just want to check that.

Let $A$ be an Abelian group of prime power order. It can be expressed as a (unique) direct product of cyclic groups of prime power order: $A = \mathbb{Z}_{p^{n_1}} \times \cdots \times \mathbb{Z}_{p^{n_t}}$ where $p$ is a prime, and $n_1, \ldots, n_t $ are positive integers. Let $B$ be a subgroup of $A$. Question Is […]

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