Math Problem Statement
Let G be a group of order 45. 1. Determine the number of Sylow 3-subgroups of G.
Solution
To determine the number of Sylow 3-subgroups of a group of order 45, we'll follow the steps using Sylow's theorems.
Step 1: Factorize the order of the group
The order of the group is given by . First, we factorize 45:
Step 2: Use Sylow's theorems
According to Sylow's theorems:
- The number of Sylow- subgroups (where is a prime divisor of the order of the group) divides the order of the group and is congruent to 1 modulo .
- Let denote the number of Sylow- subgroups.
For Sylow-3 subgroups:
- , and the number of Sylow-3 subgroups divides .
- Additionally, .
Step 3: Possible values of
Since must divide 5 and satisfy , we consider the divisors of 5:
- Divisors of 5: 1, 5
We now check each:
- : (satisfies the condition)
- : (does not satisfy the condition)
Conclusion
The only value that satisfies both conditions is .
Therefore, there is exactly 1 Sylow-3 subgroup in .
Would you like further details or have any questions?
Here are 5 related questions to expand on this topic:
- How would you determine the number of Sylow-5 subgroups for a group of order 45?
- What does it imply about if it has a unique Sylow-3 subgroup?
- How does the existence of a unique Sylow subgroup affect the normality of the subgroup in ?
- What can you say about the structure of a group with a unique Sylow-3 subgroup?
- Can a group of order 45 be simple if it has a unique Sylow-3 subgroup?
Tip: A unique Sylow- subgroup in a group is always normal in that group, which has significant implications for the group's structure.
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Math Problem Analysis
Mathematical Concepts
Group Theory
Sylow Theorems
Group Orders
Subgroup Theory
Formulas
-
Theorems
Sylow's Theorems
Suitable Grade Level
Advanced Undergraduate
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