Page 107 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 8.6 Keywords

Symmetric Group: Let X be a non-empty set. Then the system (S(Xj, 0) forms a group, called the
symmetric group of X.
Permutation: A permutation f  S, is called an r-cycle (or cycle of length r) if there are r distinct
integers i , i , i , . . . , i lying between 1 and n.
1
r
2
3
8.7 Review Questions
1. Show that (S,, °) is a non-commutative group for n  3.
 1 2 3   1 2 3
(Hint: Check the  2 3 1  and  3 2 1  don’t commute.)
   
2. Write down 2 transpositions, 2 3-cycles and a 5-cycle in S .
5
3. Show that every permutation in S is a cyclic iff n < 4.
n
4. Iff = (i i , ....i)  S,, then show that f = (i i .... i i ).
-1
2
r-1
2 1
2
r
5. Iff is an r-cycle, then show that o(f) = r, i.e., f = I and f  I, if s < r.
r
5
(Hint: If f = (i , i ....i), then f(i ) = i , r (i ) = i ,....,f (i ) = i )
2
r-1
1
2
2
1
1
1
m
3
6. Express the following cycles as products of transpositions.
(a) (1 3 5) (b) (5 3 1)
(c) (2 4 5 3)
7. Write the permutation in E3(b) as a product of transpositions.
8. Show that (1 2 .... 10) = (1 2) (2 3) . . . (9 10).
9. Check that (V , °) is a normal subgroup of A .
4
4
Answers: Self Assessment
1. (b) 2. (a) 3. (a)
4. (a) 5. (d)
8.8 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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