### What does it mean for a group to split as a direct sum of subgroups

Today I shall be talking about what it means for a group to split into a direct sum. In other words, if $G=\bigoplus\limits_{i\in I} G_i$, then what does it mean for the structure of the group?

$G$ is obviously not the union $\bigcup\limits_{i\in I} G_i$. It is a much bigger set than that in general. But it contains $G_i$ as subgroups. So what? Can we write any group as the direct sum of its subgroups? No. This means that we can write any element of the group uniquely as a linear sum of elements of $G_i$‘s. This is a much stronger statement.

The uniqueness is what is important here. Otherwise, it is true in general that we can write any element of the group as a linear combination of elements in its subgroups.