Subset Product/Examples/Example 1

From ProofWiki
Jump to navigation Jump to search

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