Problem 1 - Algebra: Pure and Applied

Question

Find the order of a Sylow \(p\) -subgroup of \(G\) for the indicated \(p\) and \(G\) : \(\begin{array}{ll}\text { (a) } p=2 & G=S_{5}\end{array}\) (b) \(p=2 \quad G=A_{4}\) (c) \(p=2\) \(G=D_{6}\) (d) \(p=3 \quad G=\) any group of order 270

Step-by-Step Solution

Answer

(a) 8, (b) 4, (c) 4, (d) 27

Step 1: Review Relevant Definitions

A Sylow \( p \)-subgroup of a group \( G \) is a maximal \( p \)-subgroup contained in \( G \). The order of a Sylow \( p \)-subgroup is \( p^k \), where \( p^k \) is the highest power of \( p \) that divides the order of \( G \).

Step 2: (a) Determine Order of \(G=S_5\)

The symmetric group \( S_5 \) has order \( 5! = 120 \).

Step 3: (a) Find Highest Power of 2 Dividing 120

The prime factorization of 120 is \( 2^3 \times 3 \times 5 \). The highest power of 2 is \( 2^3 \).

Step 4: (a) Identify Order of Sylow 2-subgroup in \( S_5 \)

Thus, the order of a Sylow 2-subgroup of \( S_5 \) is \( 2^3 = 8 \).

Step 5: (b) Determine Order of \(G=A_4\)

The alternating group \( A_4 \) has order \( 4! / 2 = 12 \).

Step 6: (b) Find Highest Power of 2 Dividing 12

The prime factorization of 12 is \( 2^2 \times 3 \). The highest power of 2 is \( 2^2 \).

Step 7: (b) Identify Order of Sylow 2-subgroup in \( A_4 \)

Thus, the order of a Sylow 2-subgroup of \( A_4 \) is \( 2^2 = 4 \).

Step 8: (c) Determine Order of \(G=D_6\)

The dihedral group \( D_6 \) represents symmetries of a hexagon and has order \( 2n = 12 \).

Step 9: (c) Find Highest Power of 2 Dividing 12

The prime factorization of 12 is \( 2^2 \times 3 \). The highest power of 2 is \( 2^2 \).

Step 10: (c) Identify Order of Sylow 2-subgroup in \( D_6 \)

Thus, the order of a Sylow 2-subgroup of \( D_6 \) is \( 2^2 = 4 \).

Step 11: (d) Determine Order of Any Group of Order 270

A group of order 270 can be factorized as \( 2 imes 3^3 \times 5 \).

Step 12: (d) Find Highest Power of 3 Dividing 270

The highest power of 3 is \( 3^3 \).

Step 13: (d) Identify Order of Sylow 3-subgroup

Thus, the order of a Sylow 3-subgroup is \( 3^3 = 27 \).

Key Concepts

symmetric group S5

The symmetric group, denoted as \(S_5\), consists of all possible permutations of five elements. This means that any arrangement or reordering of five objects can be represented as a member of this group. The order of \(S_5\) is the total number of these permutations, which is calculated as \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). So, \(S_5\) has 120 different elements.

In group theory, finding a Sylow \(p\)-subgroup involves identifying subgroups with the highest power of a prime \(p\) that divides the group's order. For \(S_5\), we look at primes like 2 and find their highest powers. Here, 120 can be factorized to \(2^3 \times 3 \times 5\). This indicates that a Sylow 2-subgroup of \(S_5\) will have an order \(2^3 = 8\). It’s fascinating how these permutations reflect complex structures in algebra and are a foundational aspect of group theory.

alternating group A4

The alternating group \(A_4\) is a subgroup of the symmetric group \(S_4\). It contains all the even permutations of four elements. Doing an even permutation means that it can be done via an even number of two-element swaps.

The order of \(A_4\) is half of \(S_4\), which totals \(12\), since \(A_4\) consists only of even permutations excluding half of the permutations in \(S_4\). If we want to find a Sylow \(p\)-subgroup where \(p=2\), we need to determine divisibility in 12 by powers of 2. Factorizing 12 gives us \(2^2 \times 3\), meaning the highest power of 2 is 4. Consequently, a Sylow 2-subgroup of \(A_4\) is of order 4, offering insight into combining symmetry and even permutations within groups.

dihedral group D6

The dihedral group \(D_6\) is representative of the symmetries of a regular hexagon, consisting of both rotations and reflections. The order of \(D_6\) is \(2n = 12\), since it includes 6 rotations and 6 reflections.

To find the Sylow \(p\)-subgroup for \(p=2\), we focus on the prime factorization of 12, which is \(2^2 \times 3\). Just as with \(A_4\), \(2^2=4\) gives us the precise measure of the order of the Sylow 2-subgroup. The rich symmetry of shapes and the transformations that preserve these forms are intricately designed in dihedral groups—fundamentals within group theory that illustrate concepts of symmetry in mathematics and physics.

group theory

Group theory is a branch of mathematics that studies the algebraic structures known as groups. A group is a set of elements combined with an operation satisfying four fundamental properties: closure, associativity, identity, and invertibility. This structure is incredibly versatile and appears across various fields of mathematics and science, from solving equations to understanding symmetry in molecules.

Within group theory, Sylow theorems play a crucial role by providing a deep understanding of the subgroups contained within a larger group. They are pivotal in examining the existence and structure of subgroups of prime power order. This is where studying

• symmetric groups such as \(S_5\)
• alternating groups like \(A_4\)
• and geometric groups, including dihedral groups \(D_6\)

serve as practical examples in unraveling the elegantly complex hierarchies and symmetries groups possess. These concepts together forge tools that address both pure theoretical questions and applied mathematical problems.