Page 87 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 6.4 Summary

We have discussed here:

The definition and example of a group homomorphism.

Let f : G  G be a group homomorphism. Then f(e ) = e ,
 1 2 1 2
[f(x)] = f(x ), Im f  G , Ker f  G .
-1
-1
2
1
A homomorphism is 1-1 iff its kernel is the trivial subgroup.

The definition and examples of a group isomorphism.

Two groups are isomorphic iff they have exactly the same algebraic structure.

The composition of group homomorphisms (isomorphisms) is a group homomorphism

(isomorphism).
6.5 Keywords

Homomorphism: Homomorphism is derived from two Greek words ‘homos’, meaning ‘link’,
and ‘morphe’, meaning ‘form’.
Inclusion Map: Let H be a subgroup of a group G. Show that the map i : H  G, i(h) = h is a
homomorphism. This function is called the inclusion map.
6.6 Review Questions

1. Show that f : (R*,.)  (R, 4) : f(x) = inx, the natural logarithm of x, is a group homomorphism.
Find Ker f and Im f also.

2. Is f : (GL (R) ,)  (w*,.) : f(A) = det(A) a homomorphism? If so, obtain Ker f and Im f.
3 3
3. Define the natural homomorphism p from S to S /A . Does (1 2) E Ker p? Does (1 2) E
3
3
3
Im p?
4. Let S = [z  C| |z| = 1}.
Define f : (R, +)  (S,.) L f(x) = e , where n is a fixed positive integer. Is f a homomorphism?
Inx
If so find Ker f.
Answers: Self Assessment

1. (a) 2. (a) 3. (b)
6.7 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).

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