About Group Theory

    One of the main unifying themes in mathematics, group theory originated from concrete examples involving the study of polynomials in the early part of the 1800's. One of the founders of the subject, Galois originally considered certain permutations of the roots of a polynomial that leave the coefficients fixed. He showed that a polynomial is solvable by radicals if and only if the associated group of permutations has certain properties. The development of group theory as an abstract algebraic system has revealed that any group, no matter how abstractly defined, is essentially the same as a group of permutations. Basically a group is a set together with a single operation that satisfies certain properties: (1) there must be an identity element, (2) every element must have an inverse, and (3) the associative law must be obeyed. This is quite a general definition in the sense that the underlying set can consist of anything; for example, the set may consist of numbers, letters, permutations, matrices, etc. But wat is required for a set to be a group is the existence of an operation on the set that obeys properties (1)-(3). In general there are many possibilities for an operation on a set. For example, for a two element set there are 16 ways of defining an operation; but only one of these operations will obey properties (1)-(3) and thus make a group. In general, (binary) operations are mappings (functions) that map two elements (a pair) from the set to a unique element back in the set. In this subject, first mappings and operations are introduced before groups to give a firm understanding of what a group is. Immediately after the abstract group is introduced permutation groups are defined and explored, thus giving a concrete approach without loss of generality.

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