Question

show that the set {1,-1} is Abelian group under multiplication

Answer 1

SOLUTION

TO PROVE

The set {1,-1} is Abelian group under multiplication

PROOF

Let G = { - 1 , 1 }

We have to show that ( G, . ) is abelian group

Check for Closure

For every a , b ∈ G we have a.b ∈ G

So G is closed under multiplication

Check for Associative

For every a , b , c ∈ G we have

a.(b.c) = (a.b).c

So G is associative under multiplication

Existence of identity

Here 1 is the identity element such that

For every a ∈ G we have

a.1 = 1.a = a

Existence of inverse

Here inG

1 is inverse of 1 and - 1 is the inverse of - 1

So ( G, . ) is a group

Check for abelian

For every a , b ∈ G we have a.b = b.a

Hence G is abelian

Hence the set {1,-1} is Abelian group under multiplication

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