Lagrange's Theorem
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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 $(G, *)$ be a finite group and let $(H, *)$ 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 $(H, *)$ of a finite group $(G, *)$ must divide the number of elements in $(G, *)$. $\blacksquare$
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