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 of order 45 has a subgroup isomorphic to (also denoted ), we can use Cauchy's Theorem from group theory.
Step 1: Apply Cauchy's Theorem
Cauchy's Theorem states that if is a finite group and is a prime number dividing the order of , then contains an element of order . Moreover, the cyclic subgroup generated by this element has order .
Step 2: Determine the relevant prime factor
The order of the group is . We can factorize 45 as: So, the prime factors of 45 are 3 and 5.
Step 3: Apply Cauchy's Theorem for
Since 5 is a prime factor of 45, by Cauchy's Theorem, there exists an element of order 5. The cyclic subgroup generated by this element is , which has order 5.
Step 4: Identify the subgroup isomorphic to
The subgroup is a cyclic group of order 5. A cyclic group of order 5 is isomorphic to , because any cyclic group of prime order is isomorphic to .
Conclusion
Therefore, has a subgroup that is isomorphic to , as required.
Would you like more details or have any questions about this proof?
Here are 5 related questions:
- How does Cauchy's Theorem apply to other prime factors, like 3, in this context?
- Can a group of order 45 have a subgroup isomorphic to ?
- What are the possible structures of the group of order 45?
- Is the subgroup unique in ?
- 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
Related Recommendation
Group of Order 5: Isomorphism and Cyclic Groups
Which Group is Isomorphic to Z/5Z? Detailed Comparison of Group Structures
Sylow Theorems and Group Structure: Exploring Group of Order 45
Group Theory Problem: Number of Sylow 3-Subgroups and Element Order in Group of Order 45
Group Isomorphism: Identifying Which Group is Isomorphic to Z/5Z