Editing Group (mathematics) (section)

=== Cosets ===
{{Main|Coset}}
In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in the symmetry group of a square, once any reflection is performed, rotations alone cannot return the square to its original position, so one can think of the reflected positions of the square as all being equivalent to each other, and as inequivalent to the unreflected positions; the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup <math>H</math> determines left and right cosets, which can be thought of as translations of <math>H</math> by an arbitrary group element {{tmath|1= g }}. In symbolic terms, the ''left'' and ''right'' cosets of {{tmath|1= H }}, containing an element {{tmath|1= g }}, are

{{Block indent|left=1.6|<math>gH=\{g\cdot h\mid h\in H\}</math> and {{tmath|1= Hg=\{h\cdot g\mid h\in H\} }}, respectively.{{sfn|Lang|2005|loc=§II.4|p=41}}}}

The left cosets of any subgroup <math>H</math> form a [[Partition of a set|partition]] of {{tmath|1= G }}; that is, the [[Union (set theory)|union]] of all left cosets is equal to <math>G</math> and two left cosets are either equal or have an [[empty set|empty]] [[Intersection (set theory)|intersection]].{{sfn|Lang|2002|loc=§I.2|p=12}} The first case <math>g_1H=g_2H</math> happens [[if and only if|precisely when]] {{tmath|1= g_1^{-1}\cdot g_2\in H }}, i.e., when the two elements differ by an element of {{tmath|1= H }}. Similar considerations apply to the right cosets of {{tmath|1= H }}. The left cosets of <math>H</math> may or may not be the same as its right cosets. If they are (that is, if all <math>g</math> in <math>G</math> satisfy {{tmath|1= gH=Hg }}), then <math>H</math> is said to be a ''[[normal subgroup]]''.

In {{tmath|1= \mathrm{D}_4 }}, the group of symmetries of a square, with its subgroup <math>R</math> of rotations, the left cosets <math>gR</math> are either equal to {{tmath|1= R }}, if <math>g</math> is an element of <math>R</math> itself, or otherwise equal to <math>U=f_{\mathrm{c}}R=\{f_{\mathrm{c}},f_{\mathrm{d}},f_{\mathrm{v}},f_{\mathrm{h}}\}</math> (highlighted in green in the Cayley table of {{tmath|1= \mathrm{D}_4 }}). The subgroup <math>R</math> is normal, because <math>f_{\mathrm{c}}R=U=Rf_{\mathrm{c}}</math> and similarly for the other elements of the group. (In fact, in the case of {{tmath|1= \mathrm{D}_4 }}, the cosets generated by reflections are all equal: {{tmath|1= f_{\mathrm{h} }R=f_{\mathrm{v} }R=f_{\mathrm{d} }R=f_{\mathrm{c} }R }}.)