To prove that $a^i$ is a generator of the cyclic group $G$ (which is of order $n$) if and only if $\gcd(i, n) = 1$, we proceed step-by-step.

Given:

• $G$ is a finite cyclic group of order $n$, so $G = \langle a \rangle$, meaning every element of $G$ can be written as some power of $a$, specifically $a^0, a^1, a^2, \ldots, a^{n-1}$.
• $\gcd(i, n) = 1$, meaning that $i$ and $n$ are coprime (i.e., their greatest common divisor is 1).

We need to show that $a^i$ is a generator of $G$, which means that the powers of $a^i$ must cover every element in $G$.

Proof:

1. Order of $a^i$: The element $a^i$ is an element of $G$, and we want to find the order of $a^i$. The order of an element $a^i$ is the smallest integer $k$ such that: $(a^i)^k = e,$ where $e$ is the identity element of $G$. Since $a^i$ is a power of $a$, we have: $(a^i)^k = a^{ik}.$ Thus, we need $a^{ik} = e$, and since $a^n = e$ (because $a$ generates a group of order $n$), this implies: $ik \equiv 0 \pmod{n}.$ This means that $k$ must satisfy $ik$ being divisible by $n$. In other words, we need $k$ to be the smallest integer such that $ik$ is a multiple of $n$. Since $\gcd(i, n) = 1$, there exists a unique solution for $k$, which is $k = n$ (because $i$ and $n$ are coprime, and multiplying $i$ by $n$ gives a multiple of $n$).

   Therefore, the order of $a^i$ is $n$.

2. Conclusion: Since the order of $a^i$ is $n$, it follows that $a^i$ generates all the elements of $G$. Specifically, the elements of $G$ can be written as: $\{(a^i)^k : k = 0, 1, 2, \ldots, n-1\} = \{a^{ik} : k = 0, 1, 2, \ldots, n-1\}.$ Since $k$ runs from $0$ to $n-1$, and the powers $a^{ik}$ cover all distinct elements of $G$, $a^i$ is indeed a generator of $G$.

Thus, we have shown that if $\gcd(i, n) = 1$, then $a^i$ is a generator of $G$.

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Would you like further clarification on any part of this proof, or have any related questions?

Here are some further questions you might find interesting:

1. How do we prove that if $\gcd(i, n) = d$, then the order of $a^i$ is $\frac{n}{d}$?
2. Can we prove a similar result for infinite cyclic groups, like $\mathbb{Z}$?
3. What is the relationship between generators of cyclic groups and primitive roots in modular arithmetic?
4. How does the structure of a cyclic group relate to the divisor function in number theory?
5. What is the order of an element in a direct product of cyclic groups, and how can this relate to generators?

Tip: When dealing with generators in cyclic groups, always check if the exponent and the group's order are coprime to determine if the element is a generator!