Isomorphism
(Monday, May 3, 2010)
Definition
A mapping f: G ® G', where áG, *ñ and áG', °ñ are groups is called isomorphism if
(i) f is one-one
(ii) f is onto, and
(iii) f is homomorphism
We wrote G @ G' and call G and G' isomorphic groups or G is isomorphic to G'.
Example 1:
A function f: Z ® G, defined by f(n) = 2n for all n e Z is a homomorphism of Z into G where áG, °ñ is a group.
Solution:
Consider f(a + b) = 2a+b, for a, b e Z
= 2a . 2b
= f(a) . f(b)
\ f is homomorphism
Example 2:
Consider the additive group áZ, +ñ of all integers and the group {-1. 1}under multiplication.
Define f: Z® H by
f(n)=
Prove that f is homomorphism.
Solution:
We have to verify, f(m + n) = f(n) . f(n)
Case (i): Let m and n be both even.
\m + n is again even
\f(m + n) = 1
Also f(m).f(n)=1 (Since, f(m)=1 & f(n)=1
\f(m+n) = f(m).f(n)
Case (ii) Let one of m and n be even say m is even and n is odd.
\ m + n is odd
\ f(m + n) = -1
Also (fm) = 1 and f(n) = -1
\f(m +n) = f(m).f(n)
Case (iii) Let m and n both be odd.
\ m + n is even
\ f(m + n) = 1
Also f(m) = -1 and f(n) = -1
\f(m +n) = f(m) . f(n)
Thus from case (i), case (ii), and case (iii) we found the f(m + n) = f(m) . f(n).
\ f is a homomorphism.
Posted in Posted by waytofeed at 7:44 AM
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