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Definition (External Direct Products $G\times H$ or $G\oplus H$)
Suppose that $G$ and $H$ are both groups. The cartesian product $G\times H$ is the set $$G\times H = \{(g,h)\mid g\in G, h\in H\}.$$ The external product of $G$ and $H$ is this Cartesian product $G\times H$ together with the group operation $(g_1,h_1)\cdot(g_2,h_2)=(g_1g_2,h_1h_2)$. We'll use the notation $G\times H$ of $G\oplus H$ to talk about the external direct product of $G$ and $H$. We can also define the external direct product of $n$ groups. The definition is analagous, where instead of having two components, we now have $n$ components, and we multiply two elements of the external direct product by performing the product componentwise.
Is there a difference between the notation $G\times H$ and $G\oplus H$? Yes. The notation $G\times H$ is called the external direct product. The notation $G\oplus H$ is called the direct sum. You will probably hear me use both interchangeably in class. These two concepts are exactly the same when looking at the external direct product, or direct sum, of two groups (or finitely many groups). The difference shows up when we consider the external product, or direct sum, of an infinite number of groups, in which case $\ds \prod G_i\neq \bigoplus G_i$. We will restrict ourselves to a finite product of groups, and for this reason we are free to use either notation as $G\times H=G\oplus H$. Feel free to read more on Wikipedia.
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