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Homework Statement
Let ##x,y,z## be arbitrary integers and the GCD: greatest common divider between any two is ##1##. Let ##(G,\cdot )## be abelian. Prove the implication:
[tex]
a,b\in G\left (a^x = b^y = (ab)^z = e\Longrightarrow a=b=e\right )
[/tex]
where ##e## is the unit element of ##G##.
Homework Equations
Corollary: For every integers ##x,y## there exist integers ##u,v## such that ##xu+yv = GCD(x,y)##
The Attempt at a Solution
If we look at what ##b## is equal to:
[tex]
b^y = e \Leftrightarrow b^yb^{1-y} = eb^{1-y} \Leftrightarrow b^1 = b^{1-y}
[/tex]
As ##x,y,z## have no common dividers, they have to be nonzero otherwise, for instance ##GCD(x,0) = x##, which is not necessarely ##1##.
At this point I'm stuck. If ##b^1 = b^{1-y}## and provided also ##y\neq 0##, then ##b\in G## is equal to one of its powers. Cyclicity?