Editing Group (mathematics) (section)

=== Subgroups ===
{{Main|Subgroup}}
Informally, a ''subgroup'' is a group <math>H</math> contained within a bigger one, {{tmath|1= G }}: it has a subset of the elements of {{tmath|1= G }}, with the same operation.{{sfn|Lang|2005|loc=§II.1|p=19}} Concretely, this means that the identity element of <math>G</math> must be contained in {{tmath|1= H }}, and whenever <math>h_1</math> and <math>h_2</math> are both in {{tmath|1= H }}, then so are <math>h_1\cdot h_2</math> and {{tmath|1= h_1^{-1} }}, so the elements of {{tmath|1= H }}, equipped with the group operation on <math>G</math> restricted to {{tmath|1= H }}, indeed form a group. In this case, the inclusion map <math>H \to G</math> is a homomorphism.

In the example of symmetries of a square, the identity and the rotations constitute a subgroup {{tmath|1= R=\{\mathrm{id},r_1,r_2,r_3\} }}, highlighted in red in the Cayley table of the example: any two rotations composed are still a rotation, and a rotation can be undone by (i.e., is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270°. The [[subgroup test]] provides a [[Necessary and sufficient conditions|necessary and sufficient condition]] for a nonempty subset {{tmath|1= H }} of a group {{tmath|1= G }} to be a subgroup: it is sufficient to check that <math>g^{-1}\cdot h\in H</math> for all elements <math>g</math> and <math>h</math> in {{tmath|1= H }}. Knowing a group's [[lattice of subgroups|subgroups]] is important in understanding the group as a whole.{{efn|However, a group is not determined by its lattice of subgroups. See {{harvnb|Suzuki|1951}}.}}

Given any subset <math>S</math> of a group {{tmath|1= G }}, the subgroup [[Generating set of a group|generated]] by <math>S</math> consists of all products of elements of <math>S</math> and their inverses. It is the smallest subgroup of <math>G</math> containing {{tmath|1= S }}.{{sfn|Ledermann|1973|loc=§II.12|p=39}} In the example of symmetries of a square, the subgroup generated by <math>r_2</math> and <math>f_{\mathrm{v}}</math> consists of these two elements, the identity element {{tmath|1= \mathrm{id} }}, and the element {{tmath|1= f_{\mathrm{h} }=f_{\mathrm{v} }\cdot r_2 }}. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup.