Types of Function :
1 ) HOMOMORPHISM OF GROUPS :
Homomorphism is a kind of function f from from one group to another.
Let (G,o) & (G’,o’) be 2 groups,
Homomorphism is a function f from from one group to another. Let (G,o) & (G’,o’) be 2 groups, a mapping/function “f ” from a group (G,o) to a group (G’,o’) is said to be a homomorphism if –
f(a o b) = f(a) o' f(b) ∀ a,b ∈ G
Example:
Q) (R,+) is a group of all real numbers under the operation ‘+’ , addition
(R -{0},*) is another group of non-zero real numbers under the operation ‘*’
(Multiplication) f is a mapping from (R,+) to (R -{0},*), defined as : f(a) = 2^a ; ∀ a ∈R
We need to verify whether f(a o b) = f(a) o' f(b)
i.e. f(a+b) = f(a).f(b)
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Homomorphism of groups :
Let (G,o) & (G’,o’) be 2 groups, a mapping “f ” from a group (G,o) to a group (G’,o’)
is said to be a homomorphism if – f(aob) = f(a) o' f(b) ∀ a,b ∈ G
Mapping “f ” is a function from one group to another
2) ISOMORPHISM OF GROUPS
Let (G,o) & (G’,o’) be 2 groups, a mapping “f ” from a group (G,o) to a group (G’,o’)
is said to be an isomorphism if –
1. f(aob) = f(a) o' f(b) ∀ a,b ∈ G
2. f is a one- one mapping
3. f is an onto mapping.
If ‘f’ is an isomorphic mapping, (G,o) will be isomorphic to the group (G’,o’) & we
write : G ≅ G'
Example 1:
Q) f(x)=log(x) for groups (R+,*) and (R,+) is a group isomorphism. Justify.
---->Solution:
1. Check for homomorphism
2. Check if one-to-one
3. Check if onto
1. Check if homomorphism :
f(x*y) = f(x)+f(y)
log(x*y) = log(x)+log(y)
so f is a homomorphism
2) Check if one-one :
f(x)=f(y) => log(x)=log(y) => x=y , so f is one-one
3) Check if onto :
f(R+)=R , so f is onto.
Let f(x)=y ⇒ loga x=y ⇒ x=a y ⇒ x=a y ∈R +
So, for every element in the co-domain, there exists some pre-image in the domain. ∴ f is
onto.
3) AUTOMORPHISM OF GROUPS :
For a group (G,+), a mapping f : G → G is called automorphism if f is one-one
f homomorphic i.e. f(a +b) = f(a) + f(b) ∀ a, b ∈ G.
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Q) f(x)=-x for group (Z,+).
-->Explanation:
as if f(a)=f(b) 🡪 -a=-b 🡪 a=b
so f is one-one.
as if f(a+b) =-(a+b) =(-a)+(-b) =f(a)+f(b),
so f is also a homomorphism.