Problem 26 - Algebra: Pure and Applied

Question

Show that in \(S_{4}\) the subgroup generated by \\{(12),(1234)\\} (in the sense of the preceding Exercise 25 ) is the whole group: \(\langle(12),(1234)\rangle=S_{4}\).

Step-by-Step Solution

Answer

The subgroup generated by \((12)\) and \((1234)\) is the whole group \(S_4\).

Step 1: Understand the Problem

We are tasked to show that the subgroup generated by \((12)\) and \((1234)\) is the whole symmetric group on 4 elements, denoted as \( S_4 \). To do this, we need to express every element of \( S_4 \) using combinations of the given cycles \((12)\) and \((1234)\).

Step 2: Identify Basic Elements in S_4

Remember, \( S_4 \) is the group of all permutations of four elements, which includes all combinations of 2-cycles, 3-cycles, and 4-cycles, in addition to the identity permutation. It has 24 elements.

Step 3: Generate 4-cycles and Identity

The 4-cycle \((1234)\) is given. Its inverse \((1432)\) can be generated by applying \((1234)\) three times, since \((1234)^4 = e\), where \(e\) is the identity. We have: \((1234)^0 = e\), \((1234)^1 = (1234)\), \((1234)^2 = (13)(24)\), \((1234)^3 = (1432)\).

Step 4: Generate Transpositions and 3-cycles

Use \((12)\) and \((1234)\):- Apply \((12)\) to generate more 2-cycles like \((34)\) (from \((13)(24)\) by conjugation with \((12)\): \((12)(13)(24)(12) = (34)\)).- Generate 3-cycles: Applying \((12\cdot(1234)\)), leads to \((124)\), similar steps can generate other 3-cycles.

Step 5: Confirm Closure and Coverage

Check that combining generated permutations leads to all other permutations in \( S_4 \): - Combining transpositions forms any needed permutation.- As shown, all 3-cycles are possible.This confirms that every permutation can be formed, proving the subgroup is indeed all of \( S_4 \).

Step 6: Conclusion

Since we can express all permutations in \( S_4 \) using \((12)\) and \((1234)\), we conclude that the subgroup generated by these permutations equals the entire group \( S_4 \).

Key Concepts

Permutations

In mathematics, permutations are fundamental when we talk about arranging objects. In simple terms, a permutation is just a rearrangement of a set of elements. For instance, if you have three elements: 1, 2, and 3, one possible permutation is the sequence 2, 3, 1.

Permutations are everywhere—from solving Rubik's Cubes to shuffling cards. Mathematically, a permutation of a set maintains each element's unique identity while changing their placed order.

• Permutations are essential in analyzing arrangements in sets.
• They find applications in various disciplines such as probability, algebra, and geometry.
• Both cyclic (where elements shift positions by a fixed step) and non-cyclic permutations exist.

Understanding permutations allows us to explore many intriguing mathematical concepts, such as how groups respond to mutations in their structure.

Group Theory

Group theory is a branch of mathematics concerned with the study of groups. A group is a set combined with an operation that has specific properties: closure, associativity, identity, and invertibility.

A group helps us understand symmetry and provides a framework for solving numerous mathematical problems. For instance, the symmetric group on 4 elements, denoted as \( S_4 \), includes all permutations of four objects totalizing 24 elements.

• Each permutation in a group maps each element of a set to another, allowing no element duplication.
• An identity element exists as a part of the group, which doesn't change any element it is combined with.
• Every element in a group must have an inverse, making it crucial for solving algebraic structures.

Group theory foundationally influences fields like physics, chemistry, and cryptography through symmetry and structural analysis.

Cycle Notation

Cycle notation is a simplified way of expressing permutations by describing how elements "cycle" through positions. In this format, each cycle is a sequence enclosed in parentheses. For example, the cycle \((1234)\) indicates that 1 goes to 2, 2 goes to 3, 3 goes to 4, and 4 returns to 1.

This notation helps streamline the understanding and manipulation of permutations.

• Each number appears exactly once within each parenthesis.
• If an element maps to itself, it is omitted for simplicity.
• Consecutive cycles can denote disjoint operations, like \((12)(34)\), mapping 1 to 2 and 3 to 4 independently.

Using cycle notation can significantly aid in visualizing complex permutation-based computations.

Subgroups

Subgroups are smaller groups hidden within a larger group, maintaining the group's same operation and properties. Defining subgroups lets us zoom into interesting parts of a larger structure to simplify analysis.

In the context of \( S_4 \), subgroups like \( \langle(12),(1234)\rangle \) can be explored to generate different elements by combining cycles.

• A subgroup must also include the group's identity and invertibility elements.
• Every operation performed with elements of the subgroup results in another element within the subgroup.
• Generating a subgroup often involves a combination of simpler permutations or transformations.

Recognizing subgroups sheds light on the symmetries and operational possibilities within larger algebraic structures.