Closed Algebraic Structure/Examples/2^m 3^n under Multiplication

Examples of Closed Algebraic Structures

Let $S$ be the set defined as:

$S := \set {2^m 3^n: m, n \in \Z}$

Then the algebraic structure $\struct {S, \times}$ is closed.

Let $a, b \in S$ such that:

$\ds a$

$=$

$\ds 2^{m_1} 3^{n_1}$

$\ds b$

$=$

$\ds 2^{m_2} 3^{n_2}$

where $m_1, m_2, n_1, n_2 \in \Z$.

Then:

$\ds a \times b$

$=$

$\ds 2^{m_1} 3^{n_1} \times 2^{m_2} 3^{n_2}$

$\ds $

$=$

$\ds 2^{m_1} 2^{m_2} 3^{n_1} 3^{n_2}$

$\ds $

$=$

$\ds 2^{m_1 + m_2} 3^{n_1 + n_2}$

$\ds $

$=$

$\ds 2^m 3^n$

where $m = m_1 + m_2$ and $n = n_1 + n_2$

$\ds $

$\in$

$\ds S$

Definition of $S$

$\blacksquare$

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures: Exercise $6.7 \ \text {(c)}$