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Chapter 7: Problem 42

Prove that every group of order \(p^{2} q\), where \(p>q\) are primes, has a normal Sylow \(p\) -subgroup, and classify all such groups.

Short Answer

Expert verified
Every group of order \[ p^2 q \] with primes \[ p > q \] has a normal Sylow p-subgroup. These groups are either direct products or semi-direct products of their Sylow subgroups.

Step by step solution

01

- Understand Sylow Theorems

The Sylow theorems provide critical information about the number, size, and properties of p-subgroups in a group. They state that the number of Sylow p-subgroups of a group G, denoted by n_p, divides the order of the group and is congruent to 1 modulo p.
02

- Determine possible values for n_p

For a group G of order \[ p^2 q \] with primes p and q (\[ p > q \]), the number of Sylow p-subgroups \[ n_p \] must divide q and be congruent to 1 modulo p. The only divisors of q are 1 and q. Hence, n_p can only be 1 or q.
03

- Analyze values for n_p

Examine the values for \[ n_p \]:If \[ n_p = 1 \], the Sylow p-subgroup is unique and thus normal in G.If \[ n_p = q \, \] then q must be congruent to 1 modulo p, which is not possible since \[ q < p \]. Hence, \[ n_p = 1 \] is the only viable option.
04

- Normality of the Sylow p-subgroup

Since \[ n_p = 1 \], there is exactly one Sylow p-subgroup in G, and it must be normal, as the unique subgroup of a given order is always normal.
05

- Classify the groups

There are two cases:1. The group G can be a direct product of its Sylow p-subgroup (which is of order \[ p^2 \]) and Sylow q-subgroup (which is of order \[ q \]).2. The group G can also be a semi-direct product where the Sylow q-subgroup acts on the Sylow p-subgroup.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sylow p-subgroup
A Sylow p-subgroup is a maximal p-subgroup of a group G, where p is a prime number. It's a subgroup whose order is a power of p. According to the Sylow theorems, if G is a finite group and p is a prime, then the number of Sylow p-subgroups, denoted _p, has specific properties:
  • _p divides the order of G
  • _p is congruent to 1 modulo p
. For a group G of order p^2 q, with p and q being primes and p > q, these properties help determine that there's exactly one Sylow p-subgroup, making it unique and therefore normal in G. This is a crucial concept in understanding the structure and classification of groups with such orders.
Group Theory
Group theory is a branch of mathematics that studies algebraic structures known as groups. A group is a set equipped with a binary operation that combines two elements to form another element. This operation must satisfy four fundamental properties:
  • Closure
  • Associativity
  • Identity element
  • Inverse elements
. Group theory provides the language and methods used to formalize symmetry and other mathematical concepts in various fields. Particularly, it helps to understand the composition and interactions within and between groups, paving the way to solving problems such as the classification of groups of order p^2 q.
Normal Subgroup
A normal subgroup N of a group G is a subgroup that is invariant under conjugation by members of G. This means for every element n in N and every g in G, the element gng^{-1} is still in N. Equivalently, we can say N is normal if _g * N = N * g for all g in G . Normal subgroups are the building blocks for forming quotient groups and are central to many areas in group theory. In our problem, if a Sylow p-subgroup is unique, it automatically becomes a normal subgroup since there cannot be any distinct p-subgroups to map to under conjugation.
Direct Product
The direct product of two groups G and H, denoted G 脳 H, is a group formed by the Cartesian product of the underlying sets, with the group operation defined component-wise. That is, if (g1, h1) and (g2, h2) are two elements in G 脳 H, their product is (g1g2, h1h2). This structure helps in understanding and constructing new groups from known ones. For groups of order p^2 q, it means G can sometimes be expressed as the direct product of its Sylow p-subgroup and Sylow q-subgroup, revealing a specific type of structure that is straightforward to analyze.
Semi-direct Product
The semi-direct product extends the concept of the direct product by allowing a more general way to combine two groups. Let G be a group, and let H and N be subgroups of G such that H is a subgroup and N is a normal subgroup. In a semi-direct product, denoted H 鈰� N, H acts on N via automorphisms. This means the group operation is defined with a twisting that depends on how H interacts with N. Understanding semi-direct products is key to classifying groups that aren't simply direct products, providing a broader and richer structure. In the problem's context, G might be a semi-direct product where the Sylow q-subgroup acts on the Sylow p-subgroup.

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Most popular questions from this chapter

. If \(H\) and \(K\) are finite groups whose orders are relatively prime, then Aut \((H \times K)\) \(\cong \operatorname{Aut}(H) \times\) Aut \((K)\). Show that this may fail if \((|H|,|K|)>1\). (Hint. Take \(H=\) \(\left.\mathbb{Z}_{p}=K .\right)\)
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