Abstract Algebra: Theory and Applications · Problem 6.5.10

The index of a subgroup is the number of distinct cosets it creates. Here, you need to show that $\mathbb{R}$ as a subgroup of $\mathbb{C}$ (under addition) generates infinitely many cosets. The key insight is that cosets are determined by their "imaginary parts"—two complex numbers are in the same coset if and only if their difference lies in $\mathbb{R}$, which means they differ only by a real number. Since the imaginary axis is uncountable and each point gives a distinct coset, the index must be infinite.

The index of a subgroup is the number of distinct cosets it creates. Here, you need to show that $\mathbb{R}$ as a subgroup of $\mathbb{C}$ (under addition) generates infinitely many cosets. The key insight is that cosets are determined by their "imaginary parts"—two complex numbers are in the same coset if and only if their difference lies in $\mathbb{R}$, which means they differ only by a real number. Since the imaginary axis is uncountable and each point gives a distinct coset, the index must be infinite.