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 , then what does it mean for the structure of the group?
is obviously not the union . It is a much bigger set than that in general. But it contains 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 ‘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.