Download e-book for iPad: An Introduction to Abstract Algebra by Derek J. S. Robinson

By Derek J. S. Robinson

ISBN-10: 3110175444

ISBN-13: 9783110175448

This undergraduate textbook for a two-semester direction in summary algebra lightly introduces the main buildings of contemporary algebra. Robinson (University of Illinois) defines the thoughts in the back of units, teams, subgroups, teams performing on units, jewelry, vector areas, box thought, and Galois idea

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Extra resources for An Introduction to Abstract Algebra

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1) π has a unique inverse function π −1 : X → X, which is also a permutation. The set of all permutations of the set X is denoted by the symbol Sym(X), which stands for the symmetric group on X. 2). In the future for the sake of simplicity we shall usually write πσ for π σ . Of course idX , the identity function on X, is a permutation. At this point we pause to note some aspects of the set Sym(X): this set is “closed” with respect to forming inverses and composites, by which we mean that if π , σ ∈ Sym(X), then π −1 and π σ belong to Sym (X).

3)). (ii) However 34 3 Introduction to groups The reader should observe that there are r different ways to write an r-cycle since any element of the cycle can be the initial element: indeed (i1 i2 . . ir ) = (i2 i3 . . ir i1 ) = · · · = (ir i1 i2 . . ir−1 ). Disjoint permutations. , they do not both move the same element. An important fact about disjoint permutations is that they commute, in contrast to permutations in general. 2) If π and σ are disjoint permutations in Sn , then π σ = σ π .

1 = n!. Cyclic permutations. Let π ∈ Sn , so that π is a permutation of the set {1, 2, . . , n}. The support of π is defined to be the set of all i such that π(i) = i, in symbols supp(π ). Now let r be an integer satisfying 1 ≤ r ≤ n. Then π is called an r-cycle if supp(π ) = {i1 , i2 , . . , ir }, with distinct ij , where π(i1 ) = i2 , π(i2 ) = i3 , . . , π(ir−1 ) = ir and π(ir ) = i1 . So π moves the integers i1 , i2 , . . , ir anticlockwise around a circle, but fixes all other integers: often π is written in the form π = (i1 i2 .

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