Page 43 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 2.6 Keywords

Binary Operation: A binary operation on S is always closed on S, but may not be closed on a subset
of S.
Abelian Group: If (G, *) is a group, where G is a finite set consisting of n elements, then we say
that (G, *) is a Finite group of order n. If G is an infinite set, then we say that (G,*) is an infinite
group.

2.7 Review Questions

1. Obtain the identity element, if it exists, for the operations   1 2 3 4   .
 2 4 3 1


2. For x  R, obtain x (if it exists) for each of the operations  1 2 3 4 
 .
-1
 2 4 3 1
3. Show that (Q, +) and (R, +) are groups.
4. Calculate (1 3) ° (1 2) in S3.
5. Write the inverse of the following in S :
3
(a) (1 2)

(b) (1 3 2)
Show that (1 2) ° (1 3 2)  (1 2) ° (1 3 2) . (This shows that in Theorem 4(b) we can’t write
–1
-1
-1
(ab)-1 = a b .)
-1 -1
2.8 Further Readings

Books Dan Saracino: Abstract Algebra; A First Course.
Mitchell and Mitchell: An Introduction to Abstract Algebra.
John B. Fraleigh: An Introduction to Abstract Algebra (Relevant Portion).

Online links www.jmilne.org/math/CourseNotes/
www.math.niu.edu
www.maths.tcd.ie/
archives.math.utk.edu

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