Let $G$ be finite group. Let $x$ and $y$ be two elements of order a power of $q$, where $q$ is prime. Is the order of $xy$ equal to a power of $q$ (or of order 1)? thanks!
Product of two elements of order q
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finite-groups
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2I changed the title from "direct product" to "product" since that's what the question actually asks about. However, the original title suggests that you're confused about what "direct product" means, so I'm not completely sure that I made the right change here. – 2012-10-19
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1Hint: The answer is no. You can find easy examples in symmetric groups. – 2012-10-19