Examples
A simple illustration of Sylow subgroups and the Sylow theorems are the dihedral group of the n-gon, D2n. For n odd, 2 = 21 is the highest power of 2 dividing the order, and thus subgroups of order 2 are Sylow subgroups. These are the groups generated by a reflection, of which there are n, and they are all conjugate under rotations; geometrically the axes of symmetry pass through a vertex and a side. By contrast, if n is even, then 4 divides the order of the group, and these are no longer Sylow subgroups, and in fact they fall into two conjugacy classes, geometrically according to whether they pass through two vertices or two faces. These are related by an outer automorphism, which can be represented by rotation through π/n, half the minimal rotation in the dihedral group.
Read more about this topic: Sylow Theorems
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