On Sylow’s theorems
1University of Oulu, Faculty of Science, Mathematics
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Group theory is a mathematical domain where groups and their properties are studied. The evolution of group theory as an area of study is said to be the result of the parallel development of a variety of different studies in mathematics. Sylow’s Theorems were a set of theorems proved around the same time the concept of group theory was being established, in the 1870s. Sylow used permutation groups in his proofs which were then later generalized and shown to hold true for all finite groups. These theorems paved the way for more detailed study of abstract groups and they have had a remarkable impact on the progress of finite group theory. Sylow’s Theorems provide information about the number and nature of the subgroups of a given finite group. The three basic theorems discovered by Sylow are discussed in the paper in detail. Sylow’s Theorems prove the existence of Sylow p-subgroups for any prime p that divides the order of the group. They show that all Sylow p-subgroups are conjugate. And finally they indicate how one can determine the number of Sylow p-subgroups which exist. Due to the power of these theorems in finite group theory they have been proved by a large number of mathematicians in a variety of different ways, as is shown in the final chapter of the paper. The theory necessary to understand the Sylow’s Theorems is covered in the first four chapters of this paper, with Sylow’s Theorems being covered in the final chapter. Lagrange’s Theorem is the first important theorem and is found in Chapter 2. Lagrange’s Theorem links the size of the group and its subgroups. The corollary of Lagrange’s Theorem is not necessarily true and Sylow’s Theorems provided a solution to this particular issue. Cauchy’s Theorem as discussed in Chapter 4, is believed to have been the inspiration for Sylow’s Theorem of existence of the subgroups. Initially Cauchy’s Theorem made use of permutation groups as did Sylow’s Theorem. Cauchy’s Theorem states that if G is a finite group and p is a prime divisor of G then the group contains an element of order p, whilst Sylow’s Theorem generalizes the finding to show that if the group G is divisible by a prime p^n then G contains a subgroup whose order is then ^n. The final section of the paper deals with the consequences that Sylow’s Theorems have in terms of practical application to finite groups. The problem of the corollary to Lagrange’s Theorem and how Sylow’s Theorem provides a solution is also dealt with. The most important references in generating the theory needed for the paper comes from Joseph. J. Rotman: A first course in Abstract Algebra, 2nd ed. (Prentice Hall, Upper Saddle River, 2000) and I. N. Hernstein: Abstract Algebra (Prentice Hall, Upper Saddle River, 1995).
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