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