Show Function is a Group Homomorphism

Consider the groups $G = \mathbb{R}$ under addition and $H = \mathbb{R}^+$ under multiplication. Show that the function $\Phi: G \to H$ defined by $\Phi(x) = e^x$ is a group homomorphism.

In this problem, we explore the concept of group homomorphisms, which are essential in understanding the structural relationships between algebraic groups. Homomorphisms are mappings between two groups that preserve the group operation. This means that if you perform an operation on elements of the first group and then map the result into the second group, you will get the same result as mapping the elements first and then performing the operation in the second group. To prove that a function is a group homomorphism, one must verify that this preservation of structure holds true for all elements in the group.

The groups we are considering here are the real numbers under addition and the positive real numbers under multiplication. The function in question, phi of x equals e to the x, maps the additive group to the multiplicative group. Demonstrating that this function is a homomorphism involves verifying that phi of a plus b equals phi of a times phi of b for all real numbers a and b. This is where properties of exponential functions become crucial, as they provide the bridge for this group operation relationship.

Understanding group homomorphisms enhances your ability to discern deeper symmetries within algebraic structures, helping to simplify complex problems by recognizing fundamentally similar spaces. This problem is a great example of how seemingly different algebraic structures (addition and multiplication) can be deeply interconnected through continuous transformations like exponentiation.