Editing Direct sum of groups (section)

== Properties ==
If {{nowrap|1=''G'' = ''H'' + ''K''}}, then it can be proven that:

* for all ''h'' in ''H'', ''k'' in ''K'', we have that {{nowrap|1=''h'' ∗ ''k'' = ''k'' ∗ ''h''}}
* for all ''g'' in ''G'', there exists unique ''h'' in ''H'', ''k'' in ''K'' such that {{nowrap|1=''g'' = ''h'' ∗ ''k''}}
* There is a cancellation of the sum in a quotient; so that {{nowrap|(''H'' + ''K'')/''K''}} is isomorphic to ''H''

The above assertions can be generalized to the case of {{nowrap|1=''G'' = Σ''H''<sub>''i''</sub>}}, where {''H''<sub>i</sub>} is a finite set of subgroups:

* if {{nowrap|''i'' ≠ ''j''}}, then for all ''h''<sub>''i''</sub> in ''H''<sub>''i''</sub>, ''h''<sub>''j''</sub> in ''H''<sub>''j''</sub>, we have that {{nowrap|1=''h''<sub>''i''</sub> ∗ ''h''<sub>''j''</sub> = ''h''<sub>''j''</sub> ∗ ''h''<sub>''i''</sub>}}
* for each ''g'' in ''G'', there exists a unique set of elements ''h''<sub>''i''</sub> in ''H''<sub>''i''</sub> such that
:''g'' = ''h''<sub>1</sub> ∗ ''h''<sub>2</sub> ∗ ... ∗ ''h''<sub>''i''</sub> ∗ ... ∗ ''h''<sub>''n''</sub>
* There is a cancellation of the sum in a quotient; so that {{nowrap|((Σ''H''<sub>''i''</sub>) + ''K'')/''K''}} is isomorphic to Σ''H''<sub>''i''</sub>.

Note the similarity with the [[direct product of groups|direct product]], where each ''g'' can be expressed uniquely as
:''g'' = (''h''<sub>1</sub>,''h''<sub>2</sub>, ..., ''h''<sub>''i''</sub>, ..., ''h''<sub>''n''</sub>).

Since {{nowrap|1=''h''<sub>''i''</sub> ∗ ''h''<sub>''j''</sub> = ''h''<sub>''j''</sub> ∗ ''h''<sub>''i''</sub>}} for all {{nowrap|''i'' ≠ ''j''}}, it follows that multiplication of elements in a direct sum is isomorphic to multiplication of the corresponding elements in the direct product; thus for finite sets of subgroups, Σ''H''<sub>''i''</sub> is isomorphic to the direct product ×{''H''<sub>''i''</sub>}.