Nonsimplicity criteria for orders of finite groups
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In this thesis some methods are developed for testing orders of groups for nonsimplicity. So far no one has been able to determine all finite simple groups, but many methods have been found which show that groups of certain orders cannot be simple. In 1963, J. Thompson and W. Feit published a proof that all groups of odd order are solvable. This implies that groups of odd order cannot be simple unless the order is a prime. It is easily seen that groups of prime order are always simple. These two facts are assumed throughout the thesis. Chapter I deals with the Sylow theorems and those results which follow directly from them. The Sylow theorems prove the existence of subgroups whose orders are all the prime powers that divide the order of the group. They also show how to determine, to a certain extent, the number of subgroups of certain orders that the group contains. By direct application of the Sylow theorems it can be proved, for example, that groups of prime power order (>p^2) cannot be simple, and that groups whose orders have only two or three prime factors cannot be simple. The main result of Chapter II is Burnside's Theorem. This theorem has a number of applications, some of which are given at the end of the chapter. For instance, it is proved that groups of order p^2q^2 and p^2q^3 where p < q (p and q are distinct prtmes) cannot be simple. Also there follows from this theorem a very general result. The order of any stmple group must be divisible by 8 or 12. Permutation groups are discussed 1n Chapter III. Here it is seen that many useful techniques for testing specific groups for nonsimplicity follow from permutation representations. Chapter IV deals with group representations. The main result is a theorem of Burnside, that all groups of order p^(aplpha)q^(beta) are solvable. This implies that they cannot be simple unless they are of prime order. It is also shown here that some of the theory developed can be applied in testing specific groups whose orders are not of the form p^(aplpha)q^(beta) for nonsimplicity. Some results that follow from the transfer are proved in Chapter V, the main one being that the order of a simple group must be divisible by 12, 16 or 56. This is the most general theorem proved in the thesis. It indicates that only those groups whose orders are multiples of 12, 16 or 56 need be treated with more specialized techniques.
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