By Derek J. S. Robinson

ISBN-10: 3110175444

ISBN-13: 9783110175448

This undergraduate textbook for a two-semester path 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 conception, and Galois conception

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**Sample text**

M2 −1 (i2 )) present in π . This is disjoint from the first cycle. Indeed, if the cycles had a common element, they would clearly have to coincide. It should also be clear that by a finite number of applications of this procedure we can express π as a product of disjoint cycles. Now to establish uniqueness. Assume that there are two expressions for π as a product of disjoint cycles, say (i1 i2 . . )(j1 j2 . . ) . . and (i1 i2 . . )(j1 j2 . . ) . . 2) disjoint cycles commute. Thus without loss of generality we can assume that i1 occurs in the cycle (i1 i2 .

If b > 0, then a − b(q + 1) = r − b < r, while if b < 0, then a − b(q − 1) = r + b < r. In each case a contradiction is reached since we have produced an integer in S which is less than r. Hence r < |b|. Finally, we must show that q and r are unique. Suppose that a = bq + r where q , r ∈ Z and 0 ≤ r < |b|. Then bq + r = bq + r and b(q − q ) = r − r. Thus |b| · |q − q | = |r − r |. If q = q , then |r − r | ≥ |b|, whereas |r − r | < |b| since 0 ≤ r, r < |b|. Therefore q = q and it follows at once that r = r .

If b > 0, then a − b(q + 1) = r − b < r, while if b < 0, then a − b(q − 1) = r + b < r. In each case a contradiction is reached since we have produced an integer in S which is less than r. Hence r < |b|. Finally, we must show that q and r are unique. Suppose that a = bq + r where q , r ∈ Z and 0 ≤ r < |b|. Then bq + r = bq + r and b(q − q ) = r − r. Thus |b| · |q − q | = |r − r |. If q = q , then |r − r | ≥ |b|, whereas |r − r | < |b| since 0 ≤ r, r < |b|. Therefore q = q and it follows at once that r = r .

### An Introduction to Abstract Algebra by Derek J. S. Robinson

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