Cyclic group, rotational symmetry group for an object with n-fold symmetry in a plane of rotation
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The Feit–Thompson theorem, that infinite simple groups that are not cyclic groups have even order, was based on the hypothesis of some, and therefore some minimal, simple group G of odd order.
Lenstra showed that a cyclic group does not have a generic polynomial if n is divisible by eight, and Smith explicitly constructs such a polynomial in case n is not divisible by eight.