Group Actions, First Sylow Theorem

First Sylow Theorem

Let |G| = mpⁿ where n is positive and p does not divide m. For each i in [1,n] there is a subgroup of order pⁱ, and if i < n, each subgroup of order pⁱ is normal in a subgroup of order pⁱ⁺¹.

To start, Macay's theorem gives us Z_p. By induction assume H has order pⁱ, where i < n.

Let K be the normalizer of H in G. We haven't reached i = n yet, so p divides the index of H in G. Use the congruent index principle (previous section) to show that K properly contains H. We know that H is normal in K; any subgroup is normal in its normalizer. The congruent index principle tells us p divides the index of H in K. By Macay's theorem, the quotient group K/H contains Z_p. Take its preimage in K to obtain a subgroup J in K. Everything in J conjugates H onto itself, since J is contained in K, hence H is normal in J. Furthermore, |J| = pⁱ⁺¹. This completes the inductive step.

The sylow subgroups are the subgroups with order pⁿ, The maximal p subgroups in G. Conjugate any one of these sylow subgroups and obtain another group of order pⁿ, another sylow subgroup. If there is but one sylow subgroup it is normal in G.

Every P-Group Has a Normal Subgroup of Index P

If G is a p-group, find a subgroup Z_p by Macay's theorem. Then find larger and larger subgroups, each containing the previous, by the first sylow theorem. Finally we reach G, and the previous subgroup is normal in G, with index p.