# Subset Product/Examples/Example 1

## Examples of Subset Product

Let $G$ be a group.

Let $a \in G$ be an element of $G$.

Let:

 $\displaystyle X$ $=$ $\displaystyle \set {e, a^2}$ $\displaystyle Y$ $=$ $\displaystyle \set {e, a, a^3}$

Let $\order a = 4$.

Then:

$\card {X Y} = 4$

where $\card {\, \cdot \,}$ denotes cardinality.

## Proof

Calculating the elements of $X Y$

 $\displaystyle e Y$ $=$ $\displaystyle \set {e, a, a^3}$ Definition of Subset Product $\displaystyle a^2 Y$ $=$ $\displaystyle \set {a^2, a^3, a^5}$ Definition of Subset Product $\displaystyle \leadsto \ \$ $\displaystyle X Y$ $=$ $\displaystyle \set {e, a, a^2, a^3, a^5}$ $\displaystyle$ $=$ $\displaystyle \set {e, a, a^2, a^3}$ as $\order a = 4$: $a^5 = a$

$\blacksquare$