let G be a group. Let a,b an element of G . Prove that G is abelian iff  (ab)^-1 = a^-1b^-1.

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embizze | High School Teacher | (Level 1) Educator Emeritus

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A group is abelian if the operation has the commutative property, i.e. ab=ba for all a,b in the group.

(1) `(ab)^(-1)=a^(-1)b^(-1)=>ab=ba`

`` `(ab)^(-1)=a^(-1)b^(-1)`

`(ab)(ab)^(-1)=(ab)a^(-1)b^(-1)`

`1=aba^(-1)b^(-1)`

`1b=aba^(-1)b^(-1)b`

`b=aba^(-1)`

`ba=aba^(-1)a`

`ba=ab`

`ab=ba` by the symmetric property.`quad`

(2) `ab=ba==>(ab)^(-1)=a^(-1)b^(-1)`

`ab=ba`

`b^(-1)ab=b^(-1)ba`

`b^(-1)ab=a`

`a^(-1)b^(-1)ab=a^(-1)a`

`a^(-1)b^(-1)ab=1`

`a^(-1)b^(-1)(ab)(ab)^(-1)=1(ab)^(-1)`

`a^(-1)b^(-1)=(ab)^(-1)`

`(ab)^(-1)=a^(-1)b^(-1)`

as required.

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