Page 96 - DMTH403_ABSTRACT_ALGEBRA

Unit 7: Homomorphism Theorem

Any infinite cyclic group is isomorphic to (Z, +). Any finite cyclic group of order n is Notes

isomorphic to ( Z , +).
Let G be a group, H  G, K  G. Then H/(H K)  HK)/K.

Let G be a group, H  G, K  G, K  H, Then (G/K)/(H/K)  G/H.

The set of automorphism of a group G, Aut G, is a group with respect to the composition

of functions.
Inn G  Aut G, for any group G.

G/Z(G)  Inn G, for any group G.

7.4 Keywords

Group Homomorphism: Iff : G  G and g : G  G are two group homomorphisms, then the
2
2
1
3
composite map g . f : G  G is also a group homomorphism.
1
3
Isomorphisms: Let G and G be two groups. A homomorphism f : G  G is called an
1
1
2
2
isomorphism if f is 1-1 and onto.
7.5 Review Questions
1. Let G be a group and H  G. Show that there exists a group G and a homomorphism
1
f : G  G such that Ker f = H.
1
2. Show that the homomorphic image of a cyclic group is cyclic i.e., if G is a cyclic group and
f : G  G’ is a homomorphism, then f(G) is cyclic.
3. Show that Z = nZ, for a fixed integer n,
(Hint: Consider f : (Z, +)  (nZ, +) : f(k) = nk)
4. Is f : Z  Z : f(x) = 0 a homomorphism? An isomorphism?

Answers: Self Assessment

1. (b) 2. (a)

7.6 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

LOVELY PROFESSIONAL UNIVERSITY 89