Question

All the quotient groups are cyclic and therefore isomorphic to \(\mathbb{Z}_{n}\) for some \(n\). In each case, find this \(n\). $$ S_{4} / A_{4} $$

Short answer

The quotient group \( S_4 / A_4 \) is isomorphic to \( \mathbb{Z}_2 \).

Step-by-step solution

Understanding the Groups

The symmetric group on 4 elements, denoted by \( S_4 \), consists of all permutations of four elements. It has 24 elements as there are 4! (factorial of 4) permutations. The alternating group \( A_4 \) consists of all even permutations of four elements, and it has 12 elements because it is a subgroup of index 2 in \( S_4 \).

Determine the Order of the Quotient Group

The order of the quotient group \( S_4 / A_4 \) is given by the formula \( \frac{|S_4|}{|A_4|} \), where \(|S_4| = 24\) and \(|A_4| = 12\). Therefore, the order of \( S_4 / A_4 \) is \( \frac{24}{12} = 2 \).

Identify the Cyclic Group

A group of order 2 is cyclic and isomorphic to \( \mathbb{Z}_2 \), which is the group of integers modulo 2. Therefore, the quotient group \( S_4 / A_4 \) is isomorphic to \( \mathbb{Z}_2 \).

Conclusion

From the analysis, we conclude that \( n \) is 2, and the quotient group \( S_4 / A_4 \) is isomorphic to \( \mathbb{Z}_2 \).

Key concepts

Understanding Symmetric Groups

Symmetric groups, often represented by the symbol \( S_n \), play a crucial role in abstract algebra, especially in the context of permutations. A symmetric group \( S_n \) is defined as the set of all permutations of \( n \) elements. For example, \( S_4 \) consists of all the possible ways to rearrange 4 distinct objects. The number of elements in a symmetric group of order \( n \) is \( n! \) (n factorial). For \( S_4 \), this means 24 different permutations because \( 4! = 24 \).

• Permutations: A permutation is a rearrangement of elements. Each arrangement counts as a distinct element in the symmetric group.
• Identity Element: The permutation where all elements remain in their initial position is called the identity permutation.
• Group Operations: In symmetric groups, the operation is the composition of permutations, where two permutations are combined into a new permutation.

Symmetric groups are foundational in studying group theory, serving as examples and often being used to demonstrate more complex concepts like quotient groups, where the question \( S_4 / A_4 \) originates from.

Exploring Alternating Groups

Alternating groups, denoted by \( A_n \), are fascinating substructures of symmetric groups. They are formed by focusing on even permutations, which are permutations achievable by performing an even number of swaps of elements.

• Even Permutations: An even permutation is one that can be decomposed into an even number of transpositions (swaps). For instance, swapping elements twice results in an even permutation.
• Order of Alternating Groups: The order of \( A_n \) is \( \frac{n!}{2} \). This simplifies to 12 for \( A_4 \) because we take half of the 24 permutations in \( S_4 \).
• Subgroups: These groups are subgroups of symmetric groups and are notable for their role in defining simple groups for \( n \geq 5 \), where they become non-trivial yet cannot be further broken down.

Alternating groups are a key concept when analyzing quotient groups, as they help simplify complex group behaviors by focusing on a subset of elements, as in the \( S_4 / A_4 \) example.

Understanding Cyclic Groups

Cyclic groups are one of the simplest forms of groups in group theory. A group is called cyclic if every element can be generated by externally iterating a single element, known as a generator.

• Generators: The generator is an element from which you can reach every other element of the group by repeated application.
• Group Order: The order of a cyclic group \( \mathbb{Z}_n \) is \( n \), where the elements are integers modulo \( n \). For example, \( \mathbb{Z}_2 \) consists of two elements: \( \{0, 1\} \).
• Quotient Groups and Cyclicity: Many quotient groups are cyclic, as in the case of \( S_4 / A_4 \), which is isomorphic to the cyclic group \( \mathbb{Z}_2 \).

Cyclic groups serve as a fundamental building block in the analysis of larger and more complex groups. Recognizing a quotient group as cyclic, like \( \mathbb{Z}_2 \), is useful in understanding the group’s structure and properties.