Lagrange's Theorem

# 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 $G$.
(1)
\begin{align} \quad G = \bigcup_{g \in G} gH \end{align}
(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 $(G, \cdot)$. $\blacksquare$