Lagrange's Theorem (Group Theory)/Examples/Order of Group with Subgroups of Order 25 and 36
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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.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Cosets and Lagrange's Theorem: Exercise $11 \ \text{(ii)}$