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:

$\order {H \cap K} = 1$

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Cosets and Lagrange's Theorem: Exercise $11 \ \text{(i)}$