Abstract Algebra: Proving/Disproving |a|=|b| if |a^2|=|b^2|

[tyrannosaurus]

Homework Statement

If |a^2|=|b^2|, prove or disprove that |a|=|b|.

Homework Equations

The hint I was given is that let a be an element of order 4n+2 and let the order of b=a2

The Attempt at a Solution

I can disprove this by looking at examples, such as in the group Z20 with letting a =2 and b=4, the |a|=10 and |b|=5, but |a^2|=5 and |b^2|=5. But I know that this does not disprove it for all groups, I need a more general solution. If anyone can help me on this it would be great.

 

[Office_Shredder]

I'm going to go on a limb and say the statement is true for some groups (for example in Z₃). When they say to disprove a conjecture, all that means is find a counterexample.