Prove that if G ≠ {e} and G has no proper non-trivial subgroup, then G is finite and O(G) is a prime number.
To prove that if a group G is not the trivial group (G ≠ {e}) and has no proper non-trivial subgroups, then G is finite and the order of G (denoted as O(G)) is a prime number, you can use the following argument:
1. Start with the assumption that G is not the trivial group, so there exists an element g ≠ e in G.
2. Since G has no proper non-trivial subgroups, every non-identity element of G generates the entire group G. In other words, for any g ≠ e in G, the subgroup generated by g, denoted as <g>, is equal to G.
3. Consider the subgroup <g>. Since it generates the entire group G,
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