Subset Product/Examples/Example 1
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Examples of Subset Product
Let $G$ be a group.
Let $a \in G$ be an element of $G$.
Let:
\(\ds X\) | \(=\) | \(\ds \set {e, a^2}\) | ||||||||||||
\(\ds Y\) | \(=\) | \(\ds \set {e, a, a^3}\) |
Let $\order a = 4$.
Then:
- $\card {X Y} = 4$
where $\card {\, \cdot \,}$ denotes cardinality.
Proof
Calculating the elements of $X Y$
\(\ds e Y\) | \(=\) | \(\ds \set {e, a, a^3}\) | Definition of Subset Product | |||||||||||
\(\ds a^2 Y\) | \(=\) | \(\ds \set {a^2, a^3, a^5}\) | Definition of Subset Product | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds X Y\) | \(=\) | \(\ds \set {e, a, a^2, a^3, a^5}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {e, a, a^2, a^3}\) | as $\order a = 4$: $a^5 = a$ |
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Cosets and Lagrange's Theorem: Exercise $1 \ \text{(i)}$