Math Problem Statement

Intersection of Sylow p-subgroups of order 42 And subgroup of order 6

Solution

Given that 42 is the product of distinct prime factors 2×3×72 \times 3 \times 7, it's important to correctly identify the nature of the Sylow subgroups and their intersections in any group GG.

Sylow pp-subgroups and Their Intersections

For a finite group GG:

  1. The Sylow 2-subgroup PP has order 21=22^1 = 2.
  2. The Sylow 3-subgroup QQ has order 31=33^1 = 3.
  3. The Sylow 7-subgroup RR has order 71=77^1 = 7.

These Sylow subgroups P,Q,RP, Q, R are typically maximal pp-subgroups corresponding to each prime factor pp.

Intersecting Sylow Subgroups

The intersection of Sylow pp-subgroups for distinct primes is generally trivial, meaning they only share the identity element. This stems from the fact that Sylow subgroups for different primes do not contain common non-identity elements.

Subgroup of Order 6

A subgroup of order 6 is non-trivial and can be constructed as follows:

  • A subgroup HH of order 6 has order 2×32 \times 3.

In a group GG of order 42, such a subgroup HH can be formed as a product of Sylow subgroups PP and QQ, where PP is of order 2 and QQ is of order 3. This is possible if the group has a specific structure that allows such a combination, e.g., a semi-direct product.

Intersection of Subgroups

  1. Intersection of Sylow pp-subgroups:

    • Intersection of PP, QQ, and RR (of orders 2, 3, and 7 respectively) in GG is trivial if GG is simple and doesn't have normal subgroups. That is, PQR={e}P \cap Q \cap R = \{e\}
  2. Intersection with Subgroup of Order 6:

    • Let HH be the subgroup of order 6.
    • The elements of HH are combinations of elements from PP and QQ.
    • The intersection of HH with any Sylow pp-subgroup would typically be:
      • HPH \cap P is a Sylow 2-subgroup of HH and thus of order 2.
      • HQH \cap Q is a Sylow 3-subgroup of HH and thus of order 3.
    • The elements of HH are combinations of elements from PP and QQ, so:
      • HP=PH \cap P = P, if PHP \subseteq H.
      • HQ=QH \cap Q = Q, if QHQ \subseteq H.

In summary:

  • The intersection of the Sylow pp-subgroups P,Q,P, Q, and RR is trivial.
  • The intersection of HH (subgroup of order 6) with PP or QQ will be non-trivial, containing elements that combine their orders.

Would you like more details or have any specific questions on this topic?

Related Questions

  1. What is the intersection of all Sylow subgroups in a simple group?
  2. How do you construct a subgroup of a specific order in a given finite group?
  3. Can you explain the Sylow theorems in detail with examples?
  4. What is the significance of non-trivial intersections of subgroups?
  5. How do the normalizers of Sylow subgroups relate to their intersections?
  6. What is the role of the center of a group in the context of Sylow subgroups?
  7. Can two Sylow subgroups of different orders intersect non-trivially?
  8. How does the structure of a group influence the behavior of its Sylow subgroups?

Tip

When dealing with intersections of subgroups, always consider the order of the subgroups involved and their relative prime factors. This will help in understanding whether the intersection is trivial or non-trivial.

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Math Problem Analysis

Mathematical Concepts

Group Theory
Finite Groups
Sylow Theorems

Formulas

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Theorems

Sylow Theorems

Suitable Grade Level

Advanced Mathematics