Lagrange's Theorem

We now have all of the tools to prove a very important and astonishing theorem regarding subgroups. This theorem is known as Lagrange's theorem and will tell us that the number of elements in a subgroup of a larger group must divide the number of elements in the larger group.

Theorem 1 (Lagrange's Theorem): Let $(G, \cdot)$ be a finite group and let $(H, \cdot)$ be a subgroup. Then the number of elements in $H$ must divide the number of elements in $G$.

• Proof: From the theorem on The Set of Left (Right) Cosets of a Subgroup Partitions the Whole Group page we know that the set of left cosets of $H$ partition $G$, that is for all $g_1, g_2 \in G$ with $g_1 \neq g_2$ we have that, $g_1H \cap g_2H = \emptyset$ and:

(1)

\begin{align} \quad G = \bigcup_{g \in G} gH \end{align}

• The number of left cosets of $H$ is the index $[G : H]$ and as we saw on the The Number of Elements in a Left (Right) Coset page, the number of elements in $gH$ is equal to the number of elements in $H$. So:

(2)

\begin{align} \quad \mid G \mid = [G : H] \mid H \mid \end{align}

• Therefore $\mid H \mid$ divides $\mid G \mid$, i.e., the number of elements in any subgroup $(H, \cdot)$ of a finite group $(G, \cdot)$ must divide the number of elements in $(G, \cdot)$. $\blacksquare$