Shallan Ley

U(23): The set of integers $<$ 23 & relatively prime to 23 with multiplication $\mod$ 23.

($Z_2_3$\{0}, $\cdot_2_3$)

Is U(23) a group?
Check for:
1. Associativity of binary operation
2. Does the Identity exist in the set?
3. Does every element have a unique inverse?

Assume S = $\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22\}$

1. Associativity:

let a, b, c $\in$ $Z_2_3$\{0}
(ab)c = $n_1$ $\cdot$ 23 + $r_1$
a(bc) = $n_2$ $\cdot$ 23 + $r_2$

We know that multiplication on the integers is associative, therefore
(ab)c = a(bc) which means by definition :
($n_1$ $\cdot$ 23 + $r_1$) = ($n_2$ $\cdot$ 23 + $r_2$)

$\cdot_2_3$ is associative

2. Identity:

let a $\in$ $Z_2_3$\{0}
a $\cdot_2_3$ x = a for some x $\in$ $Z_2_3$\{0}
let x = 1
a $\cdot_2_3$ 1 = a

Identity = 1

3. Inverses:

$\{1\}$ : 1 $\cdot_2_3$ 1 = 1
$\{2\}$ : 2 $\cdot_2_3$ 12 = 1
$\{3\}$ : 3 $\cdot_2_3$ 8 = 1
$\{4\}$ : 4 $\cdot_2_3$ 6 = 1
$\{5\}$ : 5 $\cdot_2_3$ 14 =1
$\{6\}$ : 6 $\cdot_2_3$ 4 = 1
$\{7\}$ : 7 $\cdot_2_3$ 10 = 1
$\{8\}$ : 8 $\cdot_2_3$ 3 = 1
$\{9\}$ : 9 $\cdot_2_3$ 18 = 1
$\{10\}$ : 10 $\cdot_2_3$ 7 = 1
$\{11\}$ : 11 $\cdot_2_3$ 21 = 1
$\{12\}$ : 12 $\cdot_2_3$ 2 = 1
$\{13\}$ : 13 $\cdot_2_3$ 16 = 1
$\{14\}$ : 14 $\cdot_2_3$ 5 = 1
$\{15\}$ : 15 $\cdot_2_3$ 20 = 1
$\{16\}$ : 16 $\cdot_2_3$ 13 = 1
$\{17\}$ : 17 $\cdot_2_3$ 19 = 1
$\{18\}$ : 18 $\cdot_2_3$ 9 = 1
$\{19\}$ : 19 $\cdot_2_3$ 17 = 1
$\{20\}$ : 20 $\cdot_2_3$ 15 = 1
$\{21\}$ : 21 $\cdot_2_3$ 11 = 1
$\{22\}$ : 22 $\cdot_2_3$ 22 = 1

Every element has a unique inverse

Multiplication $\mod$ 23 is associative,
the identity exists in the set, and
every element has a unique element
therefore, ($Z_2_3$\{0}, $\cdot_2_3$) is in fact a group.

The cyclic subgroup of ($Z_2_3$\{0}, $\cdot_2_3$) generated by 2 is as follows:

$< 2 >$ = $\{1,2,3,4,6,8,9,12,13,16,18\}$

If we take an element that is not in this subgroup such as 5 and multiply it by every element in the group we get the coset

(5)$< 2 >$ =$\{5,7,10,11,14,15,17,19,20,21,22 \}$

Now that all the elements in the group have been used we have two cosets
of the subgroup $< 2 >$ in ($Z_2_3$\{0}, $\cdot_2_3$) :

1. $< 2 >$ = $\{1,2,3,4,6,8,9,12,13,16,18\}$ and
2. (5)$< 2 >$ =$\{5,7,10,11,14,15,17,19,20,21,22 \}$

$\to$

page revision: 56, last edited: 02 Dec 2009 07:48