Math Problem Statement

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

Solution

To prove that the group GG of order 45 has a subgroup isomorphic to Z/5Z\mathbb{Z}/5\mathbb{Z} (also denoted Z5\mathbb{Z}_5), we can use Cauchy's Theorem from group theory.

Step 1: Apply Cauchy's Theorem

Cauchy's Theorem states that if GG is a finite group and pp is a prime number dividing the order of GG, then GG contains an element of order pp. Moreover, the cyclic subgroup generated by this element has order pp.

Step 2: Determine the relevant prime factor

The order of the group GG is G=45|G| = 45. We can factorize 45 as: 45=32×545 = 3^2 \times 5 So, the prime factors of 45 are 3 and 5.

Step 3: Apply Cauchy's Theorem for p=5p = 5

Since 5 is a prime factor of 45, by Cauchy's Theorem, there exists an element gGg \in G of order 5. The cyclic subgroup generated by this element gg is g\langle g \rangle, which has order 5.

Step 4: Identify the subgroup isomorphic to Z5\mathbb{Z}_5

The subgroup g\langle g \rangle is a cyclic group of order 5. A cyclic group of order 5 is isomorphic to Z5\mathbb{Z}_5, because any cyclic group of prime order pp is isomorphic to Zp\mathbb{Z}_p.

Conclusion

Therefore, GG has a subgroup that is isomorphic to Z5\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 Z3\mathbb{Z}_3?
  3. What are the possible structures of the group GG of order 45?
  4. Is the subgroup Z5\mathbb{Z}_5 unique in GG?
  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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Group Theory
Cauchy's Theorem
Cyclic Groups
Isomorphism

Formulas

-

Theorems

Cauchy's Theorem

Suitable Grade Level

Advanced Undergraduate