Lagrange's Theorem (Group Theory)/Examples/Order of Group with 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:

$900 \divides \order G$

where $\divides$ denotes divisibility.

Proof

From Lagrange's theorem:

$25 \divides \order G$

and:

$36 \divides \order G$

We have that:

$\lcm \set {25, 36} = 900$

Hence the result.

$\blacksquare$