Equivalence of Direct Sums
The direct sum is not unique for a group; for example, in the Klein group, V4 = C2 × C2, we have that
- V4 = <(0,1)> + <(1,0)> and
- V4 = <(1,1)> + <(1,0)>.
However, it is the content of the Remak-Krull-Schmidt theorem that given a finite group G = ∑Ai = ∑Bj, where each Ai and each Bj is non-trivial and indecomposable, then the two sums are equivalent up to reordering and isomorphism of the subgroups involved.
The Remak-Krull-Schmidt theorem fails for infinite groups; so in the case of infinite G = H + K = L + M, even when all subgroups are non-trivial and indecomposable, we cannot then assume that H is isomorphic to either L or M.
Read more about this topic: Direct Sum Of Groups
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