contestada

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

Respuesta :

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

[tex]b=ebe[/tex]
[tex]b=a^3b(a^{-1})^3[/tex]
[tex]b=a^2(aba^{-1})(a^{-1})^2[/tex]
[tex]b=a^2b^2(a^{-1})^2[/tex]

Now since [tex]b^2=aba^{-1}[/tex], you have

[tex]b^2=(a^2b^2(a^{-1})^2)^2[/tex]
[tex]aba^{-1}=a^2b^4(a^{-1})^2[/tex]
[tex]b=ab^4a^{-1}[/tex]

For the same reason, you have

[tex]b^2=(ab^4a^{-1})^2[/tex]
[tex]aba^{-1}=ab^8a^{-1}[/tex]
[tex]b=b^8[/tex]

so that the order of [tex]b[/tex] is 8, not 16.

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