Non-Abelian Group: (ab)^2 = (b^2)(a^2)?

[math-chick_41]

in a non abelian group does (ab)^2 = (b^2)(a^2)?
this isn't homework, my semester is over.

 

[matt grime]

1. Just expand (ab)^2

2. Or look in the smallest non abelian group there is: S_3

 

[math-chick_41]

it expands to abab , but how come (ab)^-1 = (b^-1)(a^-1)?

 

[matt grime]

because inversion reverses the order of the elements, it is an anti-automorphism and has nothing to do with the commutativity or otherwise of the group.

we know the inverse of ab is a c such that abc=1, bc=a^{-1}, and then c=b^{-1}a^{-1}

 

[math-chick_41]

ok, thanks.

 

[HallsofIvy]

it expands to abab , but how come (ab)^-1 = (b^-1)(a^-1)?

What do you get if you multiply (ab)(b^-1a^-1)?