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A recurring theme in science and mathematics is to find the fundamental building blocks of something, then study the different ways you can put them together. For example, in chemistry, the building blocks of everyday matter are atoms. And there are different types of bonds, which describe how they can be stuck together to make molecules In number theory, you learn that the building blocks of integers are prime numbers, and that every positive whole number has a unique factorization into primes. In group theory, the fundamental building blocks of finite groups are called simple groups, and there are many ways you can piece them together to make larger, more complex groups. One way is the direct product Suppose you have any two groups, G-1 and G-2. These two groups can be finite or infinite, abelian or non-abelian. The direct product is a way to combine these two groups into a new, larger group Its the set of all pairs where the first component is from G-1, and the second com
