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
Famous quotes containing the words direct and/or sums:
“A temple, you know, was anciently “an open place without a roof,” whose walls served merely to shut out the world and direct the mind toward heaven; but a modern meeting-house shuts out the heavens, while it crowds the world into still closer quarters.”
—Henry David Thoreau (1817–1862)
“If God lived on earth, people would break his windows.”
—Jewish proverb, quoted in Claud Cockburn, Cockburn Sums Up, epigraph (1981)