Editing Subgroup (section)

{{short description|Subset of a group that forms a group itself}}
{{other uses}}
{{Group theory sidebar |Basics}}

In [[group theory]], a branch of [[mathematics]], a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. 

Formally, given a [[group (mathematics)|group]] {{mvar|G}} under a [[binary operation]]&nbsp;∗, a [[subset]] {{mvar|H}} of {{mvar|G}} is called a '''subgroup''' of {{mvar|G}} if {{mvar|H}} also forms a group under the operation&nbsp;∗. More precisely, {{mvar|H}} is a subgroup of {{mvar|G}} if the [[Restriction (mathematics)|restriction]] of ∗ to {{math|''H'' × ''H''}} is a group operation on {{mvar|H}}. This is often denoted {{math|''H'' ≤ ''G''}}, read as "{{mvar|H}} is a subgroup of {{mvar|G}}".

The '''trivial subgroup''' of any group is the subgroup {''e''} consisting of just the identity element.{{sfn|Gallian|2013|p=61}}

A '''proper subgroup''' of a group {{mvar|G}} is a subgroup {{mvar|H}} which is a [[subset|proper subset]] of {{mvar|G}} (that is, {{math|''H'' ≠ ''G''}}). This is often represented notationally by {{math|''H'' < ''G''}}, read as "{{mvar|H}} is a proper subgroup of {{mvar|G}}". Some authors also exclude the trivial group from being proper (that is, {{math|''H'' ≠ {''e''}{{0ws}}}}).{{sfn|Hungerford|1974|p=32}}{{sfn|Artin|2011|p=43}}

If {{mvar|H}} is a subgroup of {{mvar|G}}, then {{mvar|G}} is sometimes called an '''overgroup''' of {{mvar|H}}.

The same definitions apply more generally when {{mvar|G}} is an arbitrary [[semigroup]], but this article will only deal with subgroups of groups.