Lagrange's theorem

Remarks

The quotient $|G| / |H|$ is the number of left cosets of $H$ in $G$.

Going deeper

For a student

Lagrange's theorem gives a strong limit on the subgroups of a finite group. The order of each subgroup is a divisor of the order of the group.

The idea of the proof is to count cosets:

1. The left cosets $aH$ of $H$ partition $G$.
2. Each left coset has exactly $|H|$ elements, because $h \mapsto ah$ is a bijection from $H$ to $aH$.
3. If there are $k$ cosets, then $|G| = k\,|H|$.

The number $k$ is the index $[G : H]$. Thus $|G| = [G : H] \cdot |H|$.

A worked example with $G = \mathbb{Z}/12\mathbb{Z}$ and $H = \{0, 4, 8\}$:

1. $H$ is a subgroup of order $3$.
2. The cosets are $H$, $1 + H = \{1, 5, 9\}$, $2 + H = \{2, 6, 10\}$ and $3 + H = \{3, 7, 11\}$.
3. There are $4$ cosets of $3$ elements each, and $4 \cdot 3 = 12$.

Two standard consequences are that the order of each element divides $|G|$, and that a group of prime order is cyclic.

A common error is to believe the converse. A divisor $d$ of $|G|$ need not be the order of a subgroup. The alternating group $A_4$ has order $12$, but it has no subgroup of order $6$. The converse is true for some classes of groups, for example for cyclic groups and for groups of prime power order.

The theorem needs $G$ to be finite. For an infinite group the index can still be finite, as for $2\mathbb{Z}$ in $\mathbb{Z}$, but the orders are not numbers.