Alternating Groups

Definition

Alternating groups are a series of mathematical groups that consist of all the even permutations of a finite set. They are denoted as A_n, where n represents the number of elements in the set, and they play a crucial role in group theory, particularly in understanding symmetries and solving equations. These groups are important for exploring properties of polynomials and connections to Galois Theory, especially in relation to the solvability of polynomial equations by radicals.

5 Must Know Facts For Your Next Test

1. The alternating group A_n has n!/2 elements, where n! is the factorial of n, reflecting its nature as half of the symmetric group S_n.
2. A_3 is isomorphic to the cyclic group of order 3, while A_4 has 12 elements and can be represented geometrically as symmetries of a tetrahedron.
3. For n ≥ 5, A_n is a simple group, meaning it has no normal subgroups other than itself and the trivial group, which has important implications for group theory.
4. The alternating groups serve as examples in the inverse Galois problem by demonstrating which groups can be realized as Galois groups over fields.
5. The structure and properties of alternating groups have been crucial for proving results such as Abel's theorem on the impossibility of solving general quintic equations by radicals.

Review Questions

• How do alternating groups relate to the concept of even permutations and their significance in group theory?
  • Alternating groups are specifically composed of all even permutations, which are permutations that can be formed by an even number of transpositions. This definition is crucial in understanding the broader structure of symmetric groups, as it allows us to classify permutations based on their parity. The even nature of these permutations means they maintain certain algebraic properties that make them essential for studying symmetries and solving equations within group theory.
• Discuss how alternating groups provide insights into the solvability of polynomial equations by radicals through their connection with Galois groups.
  • Alternating groups play a significant role in Galois Theory by illustrating which types of symmetry among polynomial roots can be represented by Galois groups. For example, when examining the roots of certain polynomials, one can determine whether their corresponding Galois group is an alternating group, which can indicate whether the polynomial is solvable by radicals. The simple nature of A_n for n ≥ 5 further complicates matters because it shows that not all polynomials can be solved in this manner, particularly quintics and higher-degree equations.
• Evaluate the implications of A_n being simple for n ≥ 5 on the study of Galois Theory and group theory as a whole.
  • The fact that alternating groups A_n are simple for n ≥ 5 means they have no nontrivial normal subgroups. This property has profound implications in Galois Theory, as it suggests that many quintic equations cannot be solved using radicals. The simplicity indicates that any homomorphism from these groups is either trivial or onto, leading to a deeper understanding of symmetries in higher-order polynomials. This characteristic not only influences our approach to solving equations but also contributes to fundamental advancements in both abstract algebra and modern mathematical research.

Related terms

Even Permutations: Permutations that can be expressed as an even number of transpositions, or swaps of two elements in a sequence.

Symmetric Groups: Groups that consist of all possible permutations of a finite set, denoted as S_n, which includes both even and odd permutations.

Galois Group: A group that describes the symmetries of the roots of a polynomial equation, providing insights into the solvability of the equation.