Questions about Internal Direct Product Decomposition in Group Theory

Question

I'm currently on the chapter of internal direct product in group theory, and became curious about the following questions:

1. One definition of an internal direct product in my book is that for a group $G$ with $S_i$ being a subgroup of $G$ and $G=S_1\cdots S_n$, $\prod_{i=1}^nS_i \cong G$ canonically by $h(s_1,\cdots,s_n)=s_1\cdots s_n$. My question is, is it necessary to restrict to the canonical isomorphism? I tried to find an example where $\prod_{i=1}^nS_i \cong G$ only non-canonically, but I couldn't.

2. I've learned that a finitely generated Abelian group can be decomposed into cyclic groups $G=C_1\oplus \cdots \oplus C_n$ such that $C_{i\leq k}$ is of order $m_i$ with $m_1|m_2|\cdots|m_k$ and $C_{i>k}$ is infinite. My question is, is this decomposition unique in any sense? In particular,

(1) For a finite Abelian group, each $m_i$ is uniquely determined. But when the group is not cyclic, the converse of Lagrange's theorem doesn't hold in general, so I guess $G=D_1\oplus \cdots\oplus D_n$ is possible where $D_i\neq C_i$ for some $i$ (even though they are isomorphic since they are cyclic group of same order).

(2) When $G$ is infinite, since my book proved that $m_i$s are uniquely determined only for a finite group, I think the same assertion wouldn't necessarily hold in $G$. The problem is, I'm just a beginner and haven't met with infinite group other than cyclic group so I can't come up with suitable example of such a group, i.e. $G=\bigoplus_{i=1}^l D_i$ with possibly $l\neq n$ and the finite part of some $D_i$ has different order than $m_i$.

Sorry if this question was basic, but I was so curious about this. It would be grateful if there is an example I can understand; I started learning group theory this semester.

Answer 1

1. http://groupprops.subwiki.org/wiki/Internal_direct_product

   Example: Show $\mathbb{Z}_6 \cong \mathbb{Z}_3 \oplus \mathbb{Z}_2$

2. http://www.millersville.edu/~bikenaga/abstract-algebra-1/fg-abelian-groups/fg-abelian-groups.html

   Example: Let $|G| = 2^3 \cdot 3^2 \cdot 5$. Write down all nonisomorphic abelian groups