Problem 2 - Algebra: Pure and Applied

Question

Let \(P\) be a Sylow 2 -subgroup of a group \(G\) of order 20 . If \(P\) is not a normal subgroup of \(G,\) how many conjugates does \(P\) have in \(G\) ?

Step-by-Step Solution

Answer

The Sylow 2-subgroup \( P \) has 5 conjugates in \( G \).

Step 1: Sylow's Theorem Application

According to Sylow’s theorem, the number of 2-Sylow subgroups, denoted by \( n_2 \), satisfies two conditions: \( n_2 \equiv 1 \pmod{2} \) and \( n_2 \big| \frac{20}{2} = 10 \). Therefore, the possible values for \( n_2 \) are 1, 2, 5, and 10.

Step 2: Checking Normality Condition

For a Sylow 2-subgroup \( P \) to be normal in \( G \), the number of such subgroups \( n_2 \) must be 1. Since we are given that \( P \) is not normal, \( n_2 \) cannot be 1. Thus, we consider the next possible values, which are 2, 5, or 10.

Step 3: Verifying Divisibility

The only values of \( n_2 \) from the previous step that satisfy both conditions (\( n_2 \equiv 1 \pmod{2} \) and \( n_2 \mid 10 \)) are 5 and 1, but 1 is invalid due to the non-normality condition, leaving \( n_2 = 5 \) as the only possible solution.

Step 4: Conclusion on the Number of Conjugates

Since there are 5 Sylow 2-subgroups of \( G \), this means that \( P \) must have 5 conjugates in \( G \). The number of conjugates of a subgroup equals the number of Sylow \( p \)-subgroups when it is not normal.

Key Concepts

Sylow subgroups

In group theory, what we call a Sylow subgroup arises from Sylow's theorems. These focus on prime power divisors of a group's order. Specifically, a Sylow 2-subgroup is a subgroup whose order is the highest power of 2 that divides the group order. For example, if we have a group of order 20, the largest power of 2 that divides 20 is 4 (since 20 = 2^2 × 5).

• A Sylow 2-subgroup, in this case, would have an order of 4.
• Sylow subgroups are often used to study the structure of finite groups by identifying how many such subgroups exist.
• They play a critical role in understanding the possible configurations or conjugates and the emergence of normal subgroups.

Group order

The order of a group is the total number of elements within the group. In simpler terms, think of group order as the size of a set. For instance, the mentioned group has an order of 20, which is the product of its prime factors: 2 and 5.

• Prime factors decide the kind of Sylow subgroups we can derive from the group.
• The divisors of group order also determine the number of Sylow subgroups, according to Sylow's theorems.

So, in this scenario, because 20 is divisible by 4, it dictates the possible form and arrangement of Sylow 2-subgroups.

Normal subgroup

A normal subgroup is a special kind of subgroup that remains invariant under conjugation by elements of the group. When a subgroup is normal, it means that it holds a symmetric position within the larger group structure and doesn't vary with different reference points.

• Mathematically, a subgroup \( N \) of \( G \) is normal if for every element \( g \) in \( G \), the element \( gNg^{-1} = N \).
• In the exercise, because \( P \) is not normal, meaning \( n_2 \) cannot be 1, implying more complex interactions among its elements.

Conjugates

Conjugation in group theory refers to transforming a subgroup into another through the operation of another element of the group. If a subgroup isn't normal, it has several conjugates, which are essentially versions of the subgroup viewed from different perspectives.

• For Sylow 2-subgroup \( P \), the number of conjugates is equivalent to \( n_2 \), the number of Sylow subgroups.
• Thus, since in our exercise function \( n_2 \) was determined to be 5, this implies that \( P \) has 5 conjugates within \( G \).
• Conjugates help group theorists understand more about the internal symmetry and the potential for transformations within the group's structure.