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Login NowA group is a non empty set G equivalent with a binary operation *. It is denoted by (G, *)

G × G → G satisfying the following axion

- Closure: ∀ a,b ∈ G, a * b ∈ G
- Associatie: ∀ a,b,c ∈ G(a * b) * c = a * (b * c)
- Identity Element: ∀ a ∈ G, ∃ an element e ∈ G such that

a * e = a = e * a - Inverse Element: ∀ a ∈ G, ∃ a
^{-1}∈ G such thata * a^{-1}= e = a^{-1}* a

To show, set of all integerr Z froms group under addition. Let * be defined by

a * b = a + b

Now,

**1) Closure:**

Let a, b ∈ Z

then, a * b = a + b ∈ Z so it is closed

**2)**** Associative:**

Let a, b, c ∈ Z then

(a * b) * c = (a + b) * c = a + b + c

a * (b * c) = a * (b + c) = a + b + c

so, it is associative

**3) Identity Element:**

Here, O ∈ Z such that

a * 0 = a + 0 = a

0 * a = 0 + a = a

Identity Element = e = 0

**4) Ezesistence of inverse:**

Here, ∀ a ∈ Z there exists -a ∈ Z so,

a * -a = a – a = e

-a * a = -a + a = 0 = e

so, inverse element = -a

Hence, all conditions are satisfied. So, Z forms group under addition operation.

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