Group (mathematics)
Subgroups

Informally, a subgroup is a group H {\displaystyle H} contained within a bigger one, G {\displaystyle G}: it has a subset of the elements of G {\displaystyle G}, with the same operation. Concretely, this means that the identity element of G {\displaystyle G} must be contained in H {\displaystyle H}, and whenever h 1 {\displaystyle h_{1}} and h 2 {\displaystyle h_{2}} are both in H {\displaystyle H}, then so are h 1 ⋅ h 2 {\displaystyle h_{1}\cdot h_{2}} and h 1 − 1 {\displaystyle h_{1}^{-1}}, so the elements of H {\displaystyle H}, equipped with the group operation on G {\displaystyle G} restricted to H {\displaystyle H}, indeed form a group. In this case, the inclusion map H → G {\displaystyle H\to G} is a homomorphism.

In the example of symmetries of a square, the identity and the rotations constitute a subgroup R = { i d, r 1, r 2, r 3 } {\displaystyle 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 condition for a nonempty subset H {\displaystyle H} of a group G {\displaystyle G} to be a subgroup: it is sufficient to check that g − 1 ⋅ h ∈ H {\displaystyle g^{-1}\cdot h\in H} for all elements g {\displaystyle g} and h {\displaystyle h} in H {\displaystyle H}. Knowing a group's subgroups is important in understanding the group as a whole.

Given any subset S {\displaystyle S} of a group G {\displaystyle G}, the subgroup generated by S {\displaystyle S} consists of all products of elements of S {\displaystyle S} and their inverses. It is the smallest subgroup of G {\displaystyle G} containing S {\displaystyle S}. In the example of symmetries of a square, the subgroup generated by r 2 {\displaystyle r_{2}} and f v {\displaystyle f_{\mathrm {v} }} consists of these two elements, the identity element i d {\displaystyle \mathrm {id} }, and the element f h = f v ⋅ r 2 {\displaystyle 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.
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