Intersection of Subgroups is Subgroup
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Theorem
The intersection of two subgroups of a group is itself a subgroup of that group:
- $\forall H_1, H_2 \le \struct {G, \circ}: H_1 \cap H_2 \le G$
It also follows that $H_1 \cap H_2 \le H_1$ and $H_1 \cap H_2 \le H_2$.
General Result
Let $\mathbb S$ be a set of subgroups of $\struct {G, \circ}$, where $\mathbb S \ne \O$.
Then the intersection $\ds \bigcap \mathbb S$ of the elements of $\mathbb S$ is itself a subgroup of $G$.
Also, $\ds \bigcap \mathbb S$ is the largest subgroup of $\struct {G, \circ}$ contained in each element of $\mathbb S$.
Proof
Let $H = H_1 \cap H_2$ where $H_1, H_2 \le \struct {G, \circ}$.
Then:
\(\ds \) | \(\) | \(\ds a, b \in H\) | ||||||||||||
\(\ds \) | \(\leadsto\) | \(\ds a, b \in H_1 \land a, b \in H_2\) | Definition of Set Intersection | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds a \circ b^{-1} \in H_1 \land a \circ b^{-1} \in H_2\) | Group Properties | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds a \circ b^{-1} \in H\) | Definition of Set Intersection | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds H \le G\) | One-Step Subgroup Test |
As $H \subseteq H_1$ and $H \subseteq H_2$, the other results follow directly.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.2$. Subgroups: Example $94$
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$: Theorem $17$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 36.6$ Subgroups
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms