Lagrange's Theorem (Group Theory)/Examples/Intersection of Subgroups of Order 25 and 36

Examples of Use of Lagrange's Theorem
Let $G$ be a group.

Let $H$ and $K$ be subgroups of $G$ such that:
 * $\order H = 25$
 * $\order K = 36$

where $\order {\, \cdot \,}$ denotes the order of the subgroup.

Then:
 * $\order {H \cap K} = 1$

Proof
From Intersection of Subgroups is Subgroup:
 * $H \cap K \le H$

and:
 * $H \cap K \le K$

where $\le$ denotes subgrouphood.

From Lagrange's theorem:
 * $\order {H \cap K} \divides 25$

and:
 * $\order {H \cap K} \divides 36$

As $25$ and $36$ are coprime, it follows that:
 * $\order {H \cap K} = 1$