Subset Product/Examples/Example 2

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 $\order a = 6$.

Then:

$\card {X Y} = 5$

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

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 \ \ $

$\ds X Y$

$=$

$\ds \set {e, a, a^2, a^3, a^5}$

$\blacksquare$

1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Cosets and Lagrange's Theorem: Exercise $1 \ \text{(ii)}$