Define group. Show that the set of all integers Z forms group under addition operation.

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Define group. Show that the set of all integers Z forms group under addition operation.

Suresh Chand · Accepted Answer

A 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.

Define group. Show that the set of all integers Z forms group under addition operation.

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Define group. Show that the set of all integers Z forms group under addition operation.

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