Groups and Subgroups - Main Theorem

Lagrange's theorem is one of the cornerstone results in finite group theory, establishing a fundamental constraint on the structure of subgroups.

TheoremLagrange's Theorem

Let $G$ be a finite group and $H$ a subgroup of $G$ . Then the order of $H$ divides the order of $G$ : $|H| \mid |G|$

More precisely, $|G| = |H| \cdot [G:H]$ , where $[G:H]$ denotes the index of $H$ in $G$ (the number of distinct left cosets of $H$ in $G$ ).

The proof relies on the observation that left cosets partition $G$ into disjoint sets of equal size. If $g_1H, g_2H, \ldots, g_kH$ are the distinct left cosets, then: $G = g_1H \cup g_2H \cup \cdots \cup g_kH$

Each coset $g_iH$ has exactly $|H|$ elements, giving $|G| = k \cdot |H|$ where $k = [G:H]$ .

TheoremConsequences of Lagrange's Theorem

Let $G$ be a finite group. Then:

For any $g \in G$ , the order $|g|$ divides $|G|$
For any $g \in G$ , we have $g^{|G|} = e$
If $|G| = p$ is prime, then $G$ is cyclic

ExampleApplication: Groups of Prime Order

If $|G| = p$ where $p$ is prime, then $G$ has no proper subgroups. Take any $g \neq e$ in $G$ . Then $\langle g \rangle$ is a subgroup of order $|g| > 1$ . By Lagrange, $|g|$ divides $p$ , so $|g| = p$ . Thus $\langle g \rangle = G$ , proving $G$ is cyclic.

This shows that $\mathbb{Z}_p$ is, up to isomorphism, the unique group of prime order $p$ .

The converse of Lagrange's theorem is generally false. A group of order $n$ need not have a subgroup of order $d$ for every divisor $d$ of $n$ . The alternating group $A_4$ (order 12) has no subgroup of order 6, providing a counterexample.

Remark

While Lagrange's theorem constrains subgroup orders, additional structure is needed to guarantee existence. The Sylow theorems strengthen Lagrange's result by guaranteeing subgroups of prime power order. For abelian groups, the fundamental theorem ensures that subgroups exist for all divisors.

Lagrange's theorem extends naturally to arbitrary chains of subgroups. If $K \leq H \leq G$ , then: $[G:K] = [G:H] \cdot [H:K]$

This multiplicative property of indices is crucial for analyzing group structure through successive refinements. The theorem fundamentally limits the complexity of finite groups by tying the size of any subgroup to the size of the whole group.