# An Introduction to Abstract Algebra - download pdf or read online

By Derek J. S. Robinson

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

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**Example text**

7) (The Chinese Remainder Theorem) Let a1 , a2 , . . , ak and m1 , m2 , . . , mk be integers with mi > 0; assume that gcd{mi , mj } = 1 if i = j . Then there is a common solution x of the system of congruences x ≡ a1 (mod m1 ) x ≡ a2 (mod m2 ) .. . x ≡ ak (mod mk ). When k = 2, this striking result was discovered in the 1st Century AD by the Chinese mathematician Sun Tse. 7). Put m = m1 m2 . . mk and mi = m/mi . 6) there exist an integer i such that mi i ≡ 1 (mod mi ). Now put x = a1 m1 1 + · · · + ak mk k .

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 .

Finally, by (i) we have 1 = sign(id) = sign(ππ −1 ) = sign(π ) sign(π −1 ), so that sign(π −1 ) = 1/ sign(π ) = sign(π). 7) A permutation π in Sn is even (odd ) if and only if it is a product of an even (respectively odd ) number of transpositions. 6). The subset of all even permutations in Sn is a group denoted by An , which is called the alternating group of degree n. Obviously A1 = S1 . For n > 1 exactly half of the permutations in Sn are even, as the next result shows. 8) If n > 1, there are 21 n!