Title: Examples and Insights of Cyclic Groups

Examples of Cyclic Groups

1. $\{e\}$ is a cyclic group generated by $e$.

• Cyclic Group: A group that can be generated by a single element.
• Notation: $\langle e \rangle$ represents the cyclic group generated by $e$.
• Set Representation: $\{0\} +$

2. $\langle Z_2, +_2 \rangle$

• Set: $\{0, 1\}, +_2$, is a cyclic group generated by 1.
• Generator: 1 is the generator.
• Notation: $\langle 1 \rangle = Z_2$
• Calculation:
  • $2 \times 1 = 0$
  • $3 \times 1 = 1$
• Insight: Any finite group of prime order is cyclic.

3. $\{1, -1\}$ or $\langle U_2, \cdot \rangle$

• Cyclic Group: Is generated by -1.
• Notation: $\langle -1 \rangle = \{(-1)^n | n \in \mathbb{Z}\} = \{1, -1\} = U_2$
• Note: 1 is not a generator since for any power of 1 it's always 1.

4. $\langle Z_3, +_3 \rangle$

• Set: $Z_3 = \{0, 1, 2\}$, generated elements are 1 and 2.
• Properties:
  • If group order is 3 (prime), then it must be cyclic.
  • Generating Elements:
    • $\langle 1 \rangle = \{0, 1, 2\}$
    • $\langle 2 \rangle = \{0, 2, 1\}$

5. $Z_4 +_4$

• Set: $\{0, 1, 2, 3\}$
• Generating Elements:
  • $\langle 0 \rangle = \{0\}$
  • $\langle 1 \rangle = \{0, 1, 2, 3\}$
  • $\langle 2 \rangle = \{0, 2\}$
  • $\langle 3 \rangle = \{0, 3\}$
• Properties:
  • 1 and 3 are generators.
  • $Z_4$ is cyclic.

6. $V \times$

• Klein-4 Group: Not cyclic.
• Elements: $\{e, a, b, c\}$
• Generation by elements:
  • $\langle e \rangle = \{e\}$
  • $\langle a \rangle = \{e, a\}$
  • $\langle b \rangle = \{e, b\}$
  • $\langle c \rangle = \{e, c\}$
• Note: Klein-4 group cannot be generated by a single element.

Insights

• Cyclic Group Property: A group is cyclic if it can be represented by $\{g^n\}$ for some $g$, where every element can be expressed as a power of $g$.
• Generator Importance: Identifying a generator provides a compact way to express the group and its operations.

Extended readings: