Cauchy’s theorem

Cauchy's Theorem

Let $G$ be a finite group and $p$ a prime dividing $|G|$. Then $G$ contains an element of order $p$ - equivalently, a subgroup of order $p$.

Why It’s Important

A partial converse to Lagrange’s theorem. Lagrange forbids divisor sizes; Cauchy guarantees that prime divisors are realised by subgroup orders. Cauchy’s theorem is itself a stepping-stone to Sylow’s theorems, which generalise it to prime powers.

Examples

• $|G|=15=3\cdot 5$. By Cauchy, $G$ contains elements of orders $3$ and $5$, hence subgroups $C_3$ and $C_5$.
• $|G|=12=2^2\cdot 3$. By Cauchy, $G$ has elements of orders $2$ and $3$.

Standard Proof Sketch

By induction on $|G|$ using the Orbit-Stabiliser theorem:

$$|G|=|Z(G)|+\sum _i|G|/|C_G(g_i)|,$$

summed over conjugacy class representatives $g_i$ outside the centre. If $p\mid |G|$, either $p\mid |Z(G)|$ (and we win in the abelian centre, which is immediate) or some non-central centraliser $C_G(g_i)$ has $p\mid |C_G(g_i)|<|G|$, and induction applies.

Generalisation

Sylow’s theorems strengthens Cauchy: not just an element of order $p$, but a subgroup of order $p^r$ for the maximal prime-power dividing $|G|$.