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### Abstract

These theories extend the existing proof of the first Sylow theorem
(written by Florian Kammueller and L. C. Paulson) by what are often
called the second, third and fourth Sylow theorems. These theorems
state propositions about the number of Sylow p-subgroups of a group
and the fact that they are conjugate to each other. The proofs make
use of an implementation of group actions and their properties.