Define group. Show that the set of integers is not a group with respect to subtraction operation.

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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 that a * a-1 = e = a-1 * a To show, set all integer Z doesn't from the group under subscription. 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 so, it doesn't satisfy associative. Hence, all conditions aren't satisfied. So, Z doesn't form a group under addition and subtraction.

Define group. Show that the set of integers is not a group with respect to subtraction operation.

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Define group. Show that the set of integers is not a group with respect to subtraction operation.

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