Let G = (R, ) and G′ = (R , 0) denote groups of real numbers under addition and positive real numbers under multiplication respectively. Which of the following is a group homomorphism? (A) f: G → G′, f(x) = logx for x1 else 1 (B) f: G → G′, f(x) = x 3 (C) f: G′ → G f(x) = e^x (D) f: G′ → G f(x) = logx?

Question

Let G = (R, ) and G′ = (R , 0) denote groups of real numbers under addition and positive real numbers under multiplication respectively. Which of the following is a group homomorphism? (A) f: G → G′, f(x) = logx for x>1 else 1 (B) f: G → G′, f(x) = x 3 (C) f: G′ → G f(x) = e^x (D) f: G′ → G f(x) = logx?

Answer

Solution

Definition of Group Homomorphism

A group homomorphism is a function between two groups that preserves the group operation. That is, for two groups (G,*) and (H,•), a function f: G → H is a group homomorphism if and only if f(a * b) = f(a) • f(b) for all a, b in G.

Checking for Homomorphism

Option (A)

f: G → G′, f(x) = logx for x>1 else 1

Let a, b be two elements of G. We need to check if f(a+b) = f(a) * f(b)

f(a+b) = log(a+b) if a+b > 1 else 1
f(a) * f(b) = log(a) * log(b) if a,b > 1 else 1

Now, taking log on both sides of f(a) * f(b) = f(ab) we get:
log(a) * log(b) = log(ab) if a,b > 1 else 0

Since log(a) + log(b) = log(ab), the above equation is satisfied. Hence, f is a group homomorphism.

Option (B)

f: G → G′, f(x) = x^3

Let a, b be two elements of G. We need to check if f(a+b) = f(a) * f(b)

f(a+b) = (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
f(a) * f(b) = a^3 * b^3

Clearly, f(a+b) ≠ f(a) * f(b) and hence f is not a group homomorphism.

Option (C)

f: G′ → G, f(x) = e^x

Let a, b be two elements of G′. We need to check if f(ab) = f(a) + f(b)

f(ab) = e^(ab)
f(a) + f(b) = e^a + e^b

Clearly, f(ab) ≠ f(a) + f(b) and hence f is not a group homomorphism.

Option (D)

f: G′ → G, f(x) = logx

Let a, b be two elements of G′. We need to check if f(ab) = f(a) + f(b)

f(ab) = log(ab)
f(a) + f(b) = log(a) + log(b)

Now, taking log on both sides of ab = e^(log(a) + log(b)), we get:
log(ab) = log(a) + log(b)

Hence, f(ab) = f(a) + f(b) and f is a group homomorphism.

Answer

Option (A) is a group homomorphism. Option (B), (C) and (D) are not group homomorphisms.

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