The Cartesian Product of groups 📂Abstract Algebra

The Cartesian Product of groups

Definition ¹ ²

For groups $G_{1} , \cdots , G_{n}$ and elements $\displaystyle (a_{1},\cdots , a_{n}), (b_{1} , \cdots , b_{n} ) \in \prod_{i=1}^{n} G_{i}$ of their Cartesian product, $$ (a_{1},\cdots , a_{n}) (b_{1} , \cdots , b_{n} ) = (a_{1} b_{1},\cdots , a_{n} b_{n}) $$ then $\displaystyle \prod_{i=1}^{n} G_{i}$ is called the Direct Product of $G_{1} , \cdots , G_{n}$ groups.
Especially, if $G_{1}, \cdots , G_{n}$ is an Abelian group, it is denoted by $\displaystyle \bigoplus_{i=1}^{n} G_{i}$ and also referred to as a Direct Sum.
When $G_{1}$ is a subgroup of $G$, if there exists another subgroup $G_{2}$ of $G$ satisfying the following, $G_{1}$ is called a Direct Summand. $$ G = G_{1} \oplus G_{2} $$

Properties

Let’s state $G = G_{1} \oplus G_{2}$. If $H_{1}$ is a subgroup of $G_{1}$, and $H_{2}$ is a subgroup of $G_{2}$, then $H_{1}$ and $H_{2}$ can also be represented as a direct sum, and in particular, the following holds: $$ {{ G } \over { H_{1} \oplus H_{2} }} \simeq {{ G_{1} } \over { H_{1} }} \oplus {{ G_{2} } \over { H_{2} }} $$

[1]: If $H_{1} \simeq G_{1}$ and $H_{2} \simeq \left\{ 0 \right\}$ are set, $$ G / G_{1} \simeq G_{2} $$
[2]: If $H_{1} \simeq \left\{ 0 \right\}$ is set, $$ {{ G } \over { H_{2} }} \simeq G_{1} \oplus {{ G_{2} } \over { H_{2} }} $$

Explanation

While vector spaces are groups with respect to addition, groups are not vector spaces, thus the direct sum in linear algebra does not exactly match. However, for comparison to have any meaning, it should at least be a ring.

For example, the Klein four-group satisfies $V \simeq \mathbb{Z}_{2} \times \mathbb{Z}_{2}$, and if $\gcd (m , n) = 1$, then $\mathbb{Z}_{m} \times \mathbb{Z}_{n} \simeq \mathbb{Z}_{mn}$ being a cyclic group is a known theorem.

Free group

In terms of notation, for a free Abelian group, it is convenient to express that it is isomorphic to the direct sum of the integer ring $\mathbb{Z}$. For example, if $G$ is a free Abelian group of rank $3$, $G$ could be represented as follows: $$ G \simeq \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z} $$