**Cosets and Lagrange's Theorem**

Given a subgroup *H* and some *a* in G, we define the **left coset** *aH* = {*ah* : *h* in *H*}. Because *a* is invertible, the map φ : *H* → *aH* given by φ(*h*) = *ah* is a bijection. Furthermore, every element of *G* is contained in precisely one left coset of *H*; the left cosets are the equivalence classes corresponding to the equivalence relation *a*_{1} ~ *a*_{2} if and only if *a*_{1}−1*a*_{2} is in *H*. The number of left cosets of *H* is called the index of *H* in *G* and is denoted by .

Lagrange's theorem states that for a finite group *G* and a subgroup *H*,

where |*G*| and |*H*| denote the orders of *G* and *H*, respectively. In particular, the order of every subgroup of *G* (and the order of every element of *G*) must be a divisor of |*G*|.

**Right cosets** are defined analogously: *Ha* = {*ha* : *h* in *H*}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to .

If *aH* = *Ha* for every *a* in *G*, then *H* is said to be a normal subgroup. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. More generally, if *p* is the lowest prime dividing the order of a finite group *G,* then any subgroup of index *p* (if such exists) is normal.

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“To insure the adoration of a *theorem* for any length of time, faith is not enough, a police force is needed as well.”

—Albert Camus (1913–1960)