Direct sum of groups

Imagine you have two groups, let's call them Group A and Group B. Each group has its own set of elements that can be combined in different ways, like numbers. For example, in Group A you have the elements {1, 2, 3}, and in Group B you have the elements {a, b, c}.

Now, if you want to make a bigger group that has all the elements from Group A and Group B, you can create what's called a direct sum of those groups. This means, you take one element from Group A and one element from Group B, and combine them into one new element. For example, you can combine the element 1 from Group A with the element a from Group B, and get a new element (1,a).

This way, you create a new group that has all the possible combinations of elements from Group A and Group B, like (1,a), (2,a), (3,a), (1,b), (2,b), (3,b), (1,c), (2,c), and (3,c).

In other words, the direct sum of Group A and Group B is like putting your hands together - you still have your left hand and your right hand, but now you can use them together to create new things!