Respuesta :
he group g and the positive integer n = 3 satisfy the conditions given in the problem.
Let g be the 6th group isomorphic to the rotationally symmetric group of regular hexagons. Let n = 3.
Since |g| = 6, it is clear that n |g|. Split. However, the action of g on any set cannot have orbitals of magnitude 3. To see this, note that the order of each element of g is either 1, 2, or 3. If an element of g has degree 1, then its orbit under the action of g is the element itself with magnitude 1.
If the elements of g have degree 2, then the orbitals under the action of g have magnitude 2. If the elements of g have degree 3, then the orbitals under the action of g have magnitude 3. However, g only has order elements. Since it is of order 3 (that is, a rotation of 120 degrees), the effect of g cannot have an orbit of magnitude 3.
Hence, the group g and the positive integer n = 3 satisfy the conditions given in the problem.
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