How would you show that a non-cyclic group of order 10 must contain an element $r$ of order 5?
Also, is every pair of groups homomorphic?
Thanks
2026-04-02 03:17:55.1775099875
How would you show that a non-cyclic group of order 10 must contain an element of order 5?
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Let $G$ be a group of order $10$ with no element of order $5$. By Lagrange's theorem, all elements would have order $2$ (except $e$) and so $G$ would be abelian. Let $a, b \in G$ of order $2$. Then $\{e,a,b,ab\}$ would be a subgroup of order $4$, which cannot happen because $4$ does not divide $10$.