Editing Direct sum of groups (section)

==Equivalence of decompositions into direct sums==

In the decomposition of a finite group into a direct sum of indecomposable subgroups the embedding of the subgroups is not unique. For example, in the [[Klein group]] <math>V_4 \cong C_2 \times C_2</math> we have that
: <math>V_4 = \langle(0,1)\rangle + \langle(1,0)\rangle,</math> and
: <math>V_4 = \langle(1,1)\rangle + \langle(1,0)\rangle.</math>

However, the [[Remak-Krull-Schmidt theorem]] states that given a ''finite'' group ''G'' = Σ''A''<sub>''i''</sub> = Σ''B''<sub>''j''</sub>, where each ''A''<sub>''i''</sub> and each ''B''<sub>''j''</sub> is non-trivial and indecomposable, the two sums have equal terms up to reordering and isomorphism.

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 conclude that ''H'' is isomorphic to either ''L'' or ''M''.