Coset Partition in group, direct sum decomposition in vector spaces

Question

The main application of coset is to partition a group. So that I can study the entire group using these cosets. Am I correct?

Similarly in linear algebra it's the direct sum decomposition of the vector space right? Like I can study the entire vector space using these subspaces.

So ultimately do these both (i.e., direct sum in vector spaces and cosets in group theory) have the same application? Can I say they are analogues?In group theory we use cosets to study the group in parts similarly in vector spaces we use direct sum decomposition to study the vector space in parts?

Answer 1

Direct sums of vector spaces are analogous to restricted direct products of groups, not to coset partitions.

Cosets are useful because they define an equivalence relation on a group. In the special case that the cosets are of a normal subgroup, we can give the set of cosets a group structure that carries some information about the original group. But we do not generally study the original group by studying the partition "bits". We use them for counting, we use them to define actions, but we don't use them as "pieces" to study directly in order to understand the group.

A direct sum decomposition of a vector space is not a partition: two subspaces always intersect. Arbitrary direct sum decompositions are not very useful: they become useful when the subspaces have "nice" properties (e.g., they are spaneed by eigenvectors corresponding to the same eigenvalue).

No, direct sums of vector spaces are not analogous to coset partitions of groups. The better analogue is, as I mentioned above, to a "restricted direct product decomposition": given a group $G$, if we can find normal subgroups $N_i$, $i\in I$, such that:

1. $N_i\cap \langle N_j\mid j\neq i\rangle = \{e\}$;
2. $G = \langle N_i\mid i\in I\rangle$;

then $G$ is isomorphic to a restricted direct product (or "direct sum") of the $N_i$. That is what the direct sum of subspaces is directly analogous to, and not to coset decomposition.