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 GG 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 GG is given by G=45|G| = 45. First, we factorize 45:

45=32×545 = 3^2 \times 5

Step 2: Use Sylow's theorems

According to Sylow's theorems:

  1. The number of Sylow-pp subgroups (where pp is a prime divisor of the order of the group) divides the order of the group and is congruent to 1 modulo pp.
  2. Let npn_p denote the number of Sylow-pp subgroups.

For Sylow-3 subgroups:

  • p=3p = 3
  • G=45|G| = 45, and the number of Sylow-3 subgroups n3n_3 divides 4532=5\frac{45}{3^2} = 5.
  • Additionally, n31(mod3)n_3 \equiv 1 \pmod{3}.

Step 3: Possible values of n3n_3

Since n3n_3 must divide 5 and satisfy n31(mod3)n_3 \equiv 1 \pmod{3}, we consider the divisors of 5:

  • Divisors of 5: 1, 5

We now check each:

  • n3=1n_3 = 1: 11(mod3)1 \equiv 1 \pmod{3} (satisfies the condition)
  • n3=5n_3 = 5: 52(mod3)5 \equiv 2 \pmod{3} (does not satisfy the condition)

Conclusion

The only value that satisfies both conditions is n3=1n_3 = 1.

Therefore, there is exactly 1 Sylow-3 subgroup in GG.

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:

  1. How would you determine the number of Sylow-5 subgroups for a group of order 45?
  2. What does it imply about GG if it has a unique Sylow-3 subgroup?
  3. How does the existence of a unique Sylow subgroup affect the normality of the subgroup in GG?
  4. What can you say about the structure of a group with a unique Sylow-3 subgroup?
  5. Can a group of order 45 be simple if it has a unique Sylow-3 subgroup?

Tip: A unique Sylow-pp 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

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Theorems

Sylow's Theorems

Suitable Grade Level

Advanced Undergraduate