Question

Let a,b∈G, where G is a group. Then which one of the following statement is not true.

a)If a,b,ab have same order then ab=ba
b)If a^3=e,the identity element of G and aba^-1=b^2 then order of b=16
c)ab and ba have same order
d)b and aba^-1 have same order

Answer 1

Corrected: the answer is indeed (b), but the order is not 3 as I had before.


`b=ebe`

`b=a^3b(a^(-1))^3`

`b=a^2(aba^(-1))(a^(-1))^2`

`b=a^2b^2(a^(-1))^2`

Now since
`b^2=aba^(-1)`, you have


`b^2=(a^2b^2(a^(-1))^2)^2`

`aba^(-1)=a^2b^4(a^(-1))^2`

`b=ab^4a^(-1)`

For the same reason, you have


`b^2=(ab^4a^(-1))^2`

`aba^(-1)=ab^8a^(-1)`

`b=b^8`

so that the order of
`b` is 8, not 16.