**The set of integers less than 24 and relatively prime to 24 with multiplication modulo 24.**

Proof: According to the definition, two positive integers are relatively prime if their greatest common divisor is 1.

After counting the number from 0 through 24, only 8 elements are satisfy this condition.

They are {1, 5, 7, 11, 13, 17, 19, 23}, they all have the same greatest common divisor of 1 with 24.

Let the set S={1,5,7,11,13,17,19,23}

1)Associative: let a, b, c \in (S .24)

(ab)c=n_{1} .{24}+ r_{1}

a(bc)=n_{2}.{2}4+r_{2}

Since the multiplication of integer is associative, then

(ab)c=a(bc)

n_1.24+r_1 = n_2.24+r_2

2)Identity: Assume there exist an identity e in S

let a \in S, then

a\cdot_2_4 e = a

we get e = 1, therefore a\cdot_2_4 1 = a

3)Inverse: since S \in Z_2_4, we need to find each inverse element of S.

The inverse elements of S must satisfy this propoerty: a\cdot_2_4 b = 1 for a, b\in S

(1) 1\cdot_2_4 1 = 1

(2) 5\cdot_2_4 5 = 1

(3) 7\cdot_2_4 7 = 1

(4) 11\cdot_2_4 11 = 1

(5) 13\cdot_2_4 13 = 1

(6) 17\cdot_2_4 17 = 1

(7) 19\cdot_2_4 19 = 1

(8) 23\cdot_2_4 23 = 1

Each element has an unique inverse, which every element of S is also the inverse to itself.

Therefore S= ({ 1, 5, 7, 11, 13, 17, 19, 23} \cdot_2_4) is a group.