Editing Direct sum of groups (section)

== Definition ==
A [[group (mathematics)|group]] ''G'' is called the '''direct sum'''<ref name=":0" /><ref name=":1" /> of two [[subgroup]]s ''H''<sub>1</sub> and ''H''<sub>2</sub> if
* each ''H''<sub>1</sub> and ''H''<sub>2</sub> are normal subgroups of ''G'',
* the subgroups ''H''<sub>1</sub> and ''H''<sub>2</sub> have trivial intersection (i.e., having only the [[identity element]] <math>e</math> of ''G'' in common),
* ''G'' = ⟨''H''<sub>1</sub>, ''H''<sub>2</sub>⟩; in other words, ''G'' is generated by the subgroups ''H''<sub>1</sub> and ''H''<sub>2</sub>.

More generally, ''G'' is called the  direct sum of a finite set of [[subgroup]]s {''H''<sub>''i''</sub>} if
* each ''H''<sub>''i''</sub> is a [[normal subgroup]] of ''G'',
* each ''H''<sub>''i''</sub> has trivial intersection with the subgroup {{nowrap|⟨{''H''<sub>''j''</sub> : ''j'' ≠ ''i''}⟩}},
* ''G'' = ⟨{''H''<sub>''i''</sub>}⟩; in other words, ''G'' is [[generating set of a group|generated]] by the subgroups {''H''<sub>''i''</sub>}.

If ''G'' is the direct sum of subgroups ''H'' and ''K'' then we write {{nowrap|1=''G'' = ''H'' + ''K''}}, and if ''G'' is the direct sum of a set of subgroups {''H''<sub>''i''</sub>} then we often write ''G'' = Σ''H''<sub>''i''</sub>. Loosely speaking, a direct sum is [[isomorphism|isomorphic]] to a weak direct product of subgroups.