Editing Subgroup (section)

==Basic properties of subgroups==

*The [[Identity element|identity]] of a subgroup is the identity of the group: if {{mvar|G}} is a group with identity {{mvar|e<sub>G</sub>}}, and {{mvar|H}} is a subgroup of {{mvar|G}} with identity {{mvar|e<sub>H</sub>}}, then {{math|1=''e<sub>H</sub>'' = ''e<sub>G</sub>''}}.
*The [[Inverse element|inverse]] of an element in a subgroup is the inverse of the element in the group: if {{mvar|H}} is a subgroup of a group {{mvar|G}}, and {{mvar|a}} and {{mvar|b}} are elements of {{mvar|H}} such that {{math|1=''ab'' = ''ba'' = ''e<sub>H</sub>''}}, then {{math|1=''ab'' = ''ba'' = ''e<sub>G</sub>''}}.
*If {{mvar|H}} is a subgroup of {{mvar|G}}, then the inclusion map {{math|''H'' → ''G''}} sending each element {{mvar|a}} of {{mvar|H}} to itself is a [[homomorphism]].
*The [[Intersection (set theory)|intersection]] of subgroups {{mvar|A}} and {{mvar|B}} of {{mvar|G}} is again a subgroup of {{mvar|G}}.{{sfn|Jacobson|2009|p=41}} For example, the intersection of the {{mvar|x}}-axis and {{mvar|y}}-axis in {{tmath|\R^2}} under addition is the trivial subgroup.  More generally, the intersection of an arbitrary collection of subgroups of {{mvar|G}} is a subgroup of {{mvar|G}}.
*The [[Union (set theory)|union]] of subgroups {{mvar|A}} and {{mvar|B}} is a subgroup if and only if {{math|''A'' ⊆ ''B''}} or {{math|''B'' ⊆ ''A''}}.  A non-example: {{tmath|2\Z \cup 3\Z}} is not a subgroup of {{tmath|\Z,}} because 2 and 3 are elements of this subset whose sum, 5, is not in the subset.   Similarly, the union of the {{mvar|x}}-axis and the {{mvar|y}}-axis in {{tmath|\R^2}} is not a subgroup of {{tmath|\R^2.}}
*If {{mvar|S}} is a subset of {{mvar|G}}, then there exists a smallest subgroup containing {{mvar|S}}, namely the intersection of all of subgroups containing {{mvar|S}}; it is denoted by  {{math|{{angbr|''S''}}}} and is called the [[generating set of a group|subgroup generated by {{mvar|S}}]]. An element of {{mvar|G}} is in  {{math|{{angbr|''S''}}}} if and only if it is a finite product of elements of {{mvar|S}} and their inverses, possibly repeated.{{sfn|Ash|2002}}
*Every element {{mvar|a}} of a group {{mvar|G}} generates a cyclic subgroup {{math|{{angbr|''a''}}}}. If {{math|{{angbr|''a''}}}} is [[group isomorphism|isomorphic]] to {{tmath|\Z/n\Z}} ([[Integers modulo n|the integers {{math|mod ''n''}}]]) for some positive integer {{mvar|n}}, then {{mvar|n}} is the smallest positive integer for which {{math|1=''a<sup>n</sup>'' = ''e''}}, and {{mvar|n}} is called the ''order'' of {{mvar|a}}. If {{math|{{angbr|''a''}}}} is isomorphic to {{tmath|\Z,}} then {{mvar|a}} is said to have ''infinite order''.
*The subgroups of any given group form a [[complete lattice]] under inclusion, called the [[lattice of subgroups]]. (While the [[infimum]] here is the usual set-theoretic intersection, the [[supremum]] of a set of subgroups is the subgroup ''generated by'' the set-theoretic union of the subgroups, not the set-theoretic union itself.) If {{mvar|e}} is the identity of {{mvar|G}}, then the trivial group {{math|{''e''} }} is the [[partial order|minimum]] subgroup of {{mvar|G}}, while the [[partial order|maximum]] subgroup is the group {{mvar|G}} itself.

[[File:Left cosets of Z 2 in Z 8.svg|thumb|{{mvar|G}} is the group <math>\Z/8\Z,</math> the [[Integers modulo n|integers mod 8]] under addition. The subgroup {{mvar|H}} contains only 0 and 4, and is isomorphic to <math>\Z/2\Z.</math> There are four left cosets of {{mvar|H}}: {{mvar|H}} itself, {{math|1 + ''H''}}, {{math|2 + ''H''}}, and {{math|3 + ''H''}} (written using additive notation since this is an [[Abelian group|additive group]]). Together they partition the entire group {{mvar|G}} into equal-size, non-overlapping sets. The index {{math|[''G'' : ''H'']}} is 4.]]