- #1

srfriggen

- 307

- 6

## Homework Statement

Prove that the set of all rational numbers of the form 3

^{n}6

^{m}, m,n[itex]\in[/itex]Z, is a group under multiplication.

## Homework Equations

## The Attempt at a Solution

For this problem I attempted to show that the given set has 1. an Identity element, 2. each element has an inverse, 3. Closure under multiplication, and 4. Associativity.

1. The identity element is 1

2. The inverse is 3

^{-n}6

^{-m}

3. Closure: the rationals are closed under multiplication, so closure holds, i.e.

(3

^{m}6

^{n})(3

^{k}6

^{l}) = 3

^{m+k}6

^{n+l}.

4. Associativity: This is a property of the rationals and holds, i.e.

(3

^{m}6

^{n}[itex]\ast[/itex]3

^{k}6

^{l})[itex]\ast[/itex]3

^{p}6

^{q}= 3

^{m}6

^{n}[itex]\ast[/itex](3

^{k}6

^{l}[itex]\ast[/itex]3

^{p}6

^{q})

4. Associativity: