Does anyone have an intuitive explanation for this result? My textbook uses a proof by contradiction. But it just doesn't come logically to me why this holds. What would go wrong if, for example, |a|=3 and |b|=4, where b is the largest order element?
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In an abelian group, the order of $a+b$ is always the least common multiple of the orders of $a$ and $b$ (prove it!). So if $\lvert a \rvert = 3$ and $\lvert b \rvert = 4$, then $\lvert a + b \rvert = 12$, and so it is not possible that $b$ is an element of largest order in this case. – diracdeltafunk Mar 31 '24 at 00:52