It is well-known that a group cannot be written as the union of two its proper subgroups. Has anybody come across some consequences from this fact? The small one I know is that if H is a proper subgroup of G, then G is generated by the complement G-H.
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A consequence is that if a finite group $G$ has only two proper subgroups, then the group itself must be cyclic. This is seen as follows: By the stated result the group has at least one element $g$ that does not belong to either of the proper subgroups. But if there are no other proper subgroups, then the subgroup generated by $g$ cannot be a proper one, and thus must be all of $G$. This gives an(other) easy proof of the cyclicity of the group of order $pq$, where $p<q$ are primes such that $q\not\equiv 1\pmod p$. By the Sylow theorems there is only one subgroup of order $p$ and only one of order $q$, so the above result applies. |
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