Let G be a group of order 45.
Prove that G has a subgroup that is isomorphic to Z/5Z.

To prove that the group \( G \) of order 45 has a subgroup isomorphic to \( \mathbb{Z}/5\mathbb{Z} \) (also denoted \( \mathbb{Z}_5 \)), we can use **Cauchy's Theorem** from group theory.

### Step 1: Apply Cauchy's Theorem
Cauchy's Theorem states that if \( G \) is a finite group and \( p \) is a prime number dividing the order of \( G \), then \( G \) contains an element of order \( p \). Moreover, the cyclic subgroup generated by this element has order \( p \).

### Step 2: Determine the relevant prime factor
The order of the group \( G \) is \( |G| = 45 \). We can factorize 45 as:
\[
45 = 3^2 \times 5
\]
So, the prime factors of 45 are 3 and 5.

### Step 3: Apply Cauchy's Theorem for \( p = 5 \)
Since 5 is a prime factor of 45, by Cauchy's Theorem, there exists an element \( g \in G \) of order 5. The cyclic subgroup generated by this element \( g \) is \( \langle g \rangle \), which has order 5.

### Step 4: Identify the subgroup isomorphic to \( \mathbb{Z}_5 \)
The subgroup \( \langle g \rangle \) is a cyclic group of order 5. A cyclic group of order 5 is isomorphic to \( \mathbb{Z}_5 \), because any cyclic group of prime order \( p \) is isomorphic to \( \mathbb{Z}_p \).

### Conclusion
Therefore, \( G \) has a subgroup that is isomorphic to \( \mathbb{Z}_5 \), as required.

Would you like more details or have any questions about this proof?

Here are 5 related questions:
1. How does Cauchy's Theorem apply to other prime factors, like 3, in this context?
2. Can a group of order 45 have a subgroup isomorphic to \( \mathbb{Z}_3 \)?
3. What are the possible structures of the group \( G \) of order 45?
4. Is the subgroup \( \mathbb{Z}_5 \) unique in \( G \)?
5. How would the Sylow theorems apply to a group of order 45?

**Tip:** When dealing with groups of composite order, always consider the prime factors and apply Cauchy's Theorem to determine the existence of cyclic subgroups of prime order.

Learn how to prove that a group of order 45 contains a subgroup isomorphic to Z/5Z using Cauchy's Theorem and properties of cyclic groups. This problem explores fundamental concepts in group theory and is suitable for advanced undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation