Direct Product

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Formal Definition

Let $G_{1},G_{2},...,G_{n}$ be groups. For $(a_{1},a_{2},...,a_{n})$ and $(b_{1},b_{2},...,b_{n})$ in $\prod_{i=1} ^{n} G_{i}$, define $(a_{1},a_{2},...,a_{n})(b_{1},b_{2},...,b_{n})$ to be the element $(a_{1}b_{1},a_{2}b_{2},...,a_{n}b_{n})$. Then $\prod_{i=1} ^{n} G_{i}$ is a group, the direct product of the groups $G_{i}$, under the binary operation.

Informal Definition

The direct product takes two groups $G$ and $H$ and constructs a new group, usually denoted $G\times H$.

Example(s)

Let $\mathbb{R}$ be the group of real numbers under addition. The direct product, $\mathbb{R}\times\mathbb{R}$ is the group of all two component vectors under the operation of vector addition.

(1)
\begin{equation} (x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2) \end{equation}

Non-example(s)

Let $\mathbb{R}$ be a field, $\mathbb{R}\times\mathbb{R} = \{(x,y)|x,y\in\mathbb{R}\}$ would not exist because it does not result in a field since the element $(1,0)$ does not have an inverse.

Additional Comments

The direct product generalizes the cartesian product of underlying sets.

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