Intersection with Subgroup Product of Superset
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Theorem
Let $X, Y, Z$ be subgroups of a group $\struct {G, \circ}$.
Let $Y \subseteq X$.
Then:
- $X \cap \paren {Y \circ Z} = Y \circ \paren {X \cap Z}$
where $Y \circ Z$ denotes subset product.
Proof
By definition of set equality, it suffices to prove:
- $X \cap \paren {Y \circ Z} \subseteq Y \circ \paren {X \cap Z}$
and:
- $Y \circ \paren {X \cap Z} \subseteq X \cap \paren {Y \circ Z}$
$X \cap \paren {Y \circ Z}$ is contained in $Y \circ \paren {X \cap Z}$
Let $s \in X \cap \paren {Y \circ Z}$.
\(\ds s\) | \(\in\) | \(\ds X \cap \paren {Y \circ Z}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds s\) | \(\in\) | \(\ds X\) | Definition of Set Intersection | ||||||||||
\(\, \ds \land \, \) | \(\ds s\) | \(\in\) | \(\ds Y \circ Z\) | Definition of Set Intersection | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists y \in Y, z \in Z: \, \) | \(\ds s\) | \(=\) | \(\ds y \circ z\) | Definition of Subset Product | |||||||||
\(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds z\) | \(=\) | \(\ds y^{-1} \circ s\) | Group has Latin Square Property |
Then:
\(\ds Y\) | \(\subseteq\) | \(\ds X\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(\in\) | \(\ds X\) | Definition of Subset |
So we have:
\(\ds y\) | \(\in\) | \(\ds X\) | ||||||||||||
\(\, \ds \land \, \) | \(\ds s\) | \(\in\) | \(\ds X\) | Definition of Subset | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds y^{-1} \circ s\) | \(\in\) | \(\ds X\) | One-Step Subgroup Test | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds z\) | \(\in\) | \(\ds X\) | from $(1)$ |
Thus by definition of set intersection:
- $z \in X \cap Z$
So:
- $s = y \circ z \in Y \circ \paren {X \cap Z}$
By definition of subset:
- $X \cap \paren {Y \circ Z} \subseteq Y \circ \paren {X \cap Z}$
$\Box$
$Y \circ \paren {X \cap Z}$ is contained in $X \cap \paren {Y \circ Z}$
\(\ds Y \circ X\) | \(\subseteq\) | \(\ds X \circ X\) | Subset Relation is Compatible with Subset Product | |||||||||||
\(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds Y \circ X\) | \(\subseteq\) | \(\ds X\) | Product of Subgroup with Itself |
Then:
\(\ds Y \circ \paren {X \cap Z}\) | \(\subseteq\) | \(\ds \paren {Y \circ X} \cap \paren {Y \circ Z}\) | Product of Subset with Intersection | |||||||||||
\(\ds \) | \(\subseteq\) | \(\ds X \cap \paren {Y \circ Z}\) | from $(1)$ |
$\Box$
We have established that:
- $x \in Y \circ \paren {X \cap Z} \iff x \in X \cap \paren {Y \circ Z}$
From the definition of set equality:
- $Y \circ \paren {X \cap Z} = X \cap \paren {Y \circ Z}$
$\blacksquare$
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 35 \theta$