91Ó°ÊÓ

A normal subgroup is an essential concept in group theory. It is a subgroup that remains unchanged when conjugated by any member of the original group. This means if \( N \) is a normal subgroup of \( G \), then for all \( g \) in \( G \) and \( n \) in \( N \), the element \( gng^{-1} \) is also in \( N \). Normal subgroups allow us to form quotient groups, which are crucial for understanding the structure of groups.
A group of order 108 has a normal subgroup, thanks to the uniqueness of certain Sylow subgroups. When a Sylow subgroup is unique, it automatically becomes normal by the Sylow theorems, simplifying the structure of our group into more manageable parts.

Group theory is a branch of mathematics that studies the algebraic structures known as groups. A group consists of a set equipped with an operation that combines any two elements to form a third element, while satisfying four fundamental properties:

• Closure: The group operation on any two elements produces another element in the group.
• Associativity: The group operation is associative, meaning \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
• Identity: There exists an identity element that, when used in the operation with any other element, returns the same element.
• Invertibility: Every element has an inverse, so when the operation is used with its inverse, it returns the identity element.

In this context, group theory helps identify and prove the existence of subgroups, like Sylow subgroups, in larger groups, leading us to discover normal subgroups and thereby unveil the group's structure.

Sylow 2-subgroups are subgroups whose order is a power of 2. They are identified using Sylow theorems, which provide a set of rules about the number and size of these subgroups within a larger group. Specifically, the number of Sylow 2-subgroups, denoted \(n_2\), must divide \( \frac{108}{2^2} = 27 \) and also be congruent to 1 modulo 2.
This means possible values for \(n_2\) are 1, 3, 9, and 27. If there is only one Sylow 2-subgroup (\(n_2 = 1\)), it is automatically normal. However, in this case, multiple possible values mean we need to explore further to find a normal subgroup.

Sylow 3-subgroups are those whose order is a power of 3 within a group. To find these subgroups in a group of order 108, Sylow theorems again help by determining possible numbers of such subgroups. Here, the number of Sylow 3-subgroups \(n_3\) must divide \( \frac{108}{3^3} = 4 \) and be congruent to 1 modulo 3.
For the given group, this yields just one solution: \(n_3 = 1\), meaning there is a unique Sylow 3-subgroup. According to Sylow's theorems, any Sylow subgroup that is unique in its cardinality is automatically a normal subgroup. Thus, the Sylow 3-subgroup is normal in this group of order 108. This finding showcases the power of Sylow theorems in simplifying group structures by identifying these normal subgroups.