# Lagrange's Theorem (Group Theory)/Examples

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## Examples of Use of Lagrange's Theorem

### Intersection of Subgroups of Order $25$ and $36$

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:

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

### Order of Group with Subgroups of Order $25$ and $36$

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.

### Order of Union of Subgroups of Order $16$

Let $G$ be a group whose identity is $e$.

Let $H$ and $K$ be subgroups of $G$ such that:

- $\order H = \order K = 16$
- $H \ne K$

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

Then:

- $24 \le \order {H \cup K} \le 31$