Let a belong to a group and a^12 = e. express the inverse of each of the following elements in the form ak for some positive integer k. A) a^2 B) a^3 C) a^7

Given that \( a \) belongs to a group and satisfies \( a^{12} = e \) (where \( e \) is the identity element), we are to express the inverse of each of the given elements \( a^2 \), \( a^3 \), and \( a^7 \) in the form \( a^k \) for some positive integer \( k \).

### Step-by-step Solution

Since \( a^{12} = e \), this implies that the order of \( a \) divides 12. Therefore, for any integer \( m \), we have:
\[
a^{12} = e \quad \Rightarrow \quad a^{-1} = a^{12 - 1} = a^{11}
\]

#### A) Finding the inverse of \( a^2 \):

The inverse of \( a^2 \) is an element \( x \) such that:
\[
a^2 \cdot x = e
\]
Let \( x = a^k \), then:
\[
a^2 \cdot a^k = a^{2 + k} = e
\]
Thus, \( 2 + k \equiv 0 \pmod{12} \). Therefore, \( k \equiv -2 \pmod{12} \), which implies \( k = 12 - 2 = 10 \).

The inverse of \( a^2 \) is \( a^{10} \).

#### B) Finding the inverse of \( a^3 \):

The inverse of \( a^3 \) is an element \( x \) such that:
\[
a^3 \cdot x = e
\]
Let \( x = a^k \), then:
\[
a^3 \cdot a^k = a^{3 + k} = e
\]
Thus, \( 3 + k \equiv 0 \pmod{12} \). Therefore, \( k \equiv -3 \pmod{12} \), which implies \( k = 12 - 3 = 9 \).

The inverse of \( a^3 \) is \( a^{9} \).

#### C) Finding the inverse of \( a^7 \):

The inverse of \( a^7 \) is an element \( x \) such that:
\[
a^7 \cdot x = e
\]
Let \( x = a^k \), then:
\[
a^7 \cdot a^k = a^{7 + k} = e
\]
Thus, \( 7 + k \equiv 0 \pmod{12} \). Therefore, \( k \equiv -7 \pmod{12} \), which implies \( k = 12 - 7 = 5 \).

The inverse of \( a^7 \) is \( a^{5} \).

### Final Answers

- The inverse of \( a^2 \) is \( a^{10} \).
- The inverse of \( a^3 \) is \( a^{9} \).
- The inverse of \( a^7 \) is \( a^{5} \).

Would you like more details or have any questions?

### Related Questions

1. If \( a^{6} = e \), what is the inverse of \( a^4 \)?
2. Prove that \( a^{12} = e \) implies \( a^m \cdot a^{12 - m} = e \) for any integer \( m \).
3. If \( a \) is an element of order 6, what are the possible values of \( a^3 \)?
4. In a cyclic group of order 12, find the order of \( a^3 \).
5. Express the inverse of \( a^5 \) when \( a^{15} = e \).

### Tip

When working with groups, always keep in mind the group’s order properties, as they help simplify calculations with exponents modulo the order.

Let a belong to a group and a^12 = e. Express the inverse of each of the following elements in the form ak for some positive integer k. A) a^2 B) a^3 C) a^7

Learn how to find the inverses of group elements when a^12 = e. This abstract algebra problem demonstrates how to express the inverses of a^2, a^3, and a^7 in terms of powers of a using modulo arithmetic. Detailed explanations included.

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