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.

Published by ayushkhaitan3437

Hello! My name is Ayush Khaitan, and I'm a graduate student in Mathematics. I am always excited about talking to people about their research. Please please set up a meeting with me if you feel that I might have an interesting perspective to offer- https://calendly.com/ayushkhaitan/meeting-with-ayush

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