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Chapter 2: Problem 13

Prove that if a normal subgroup of \(A_{n}\) contains even a single 3 -cycle it must be all of \(A_{n}\).

Short Answer

Expert verified
If \(N\) includes any 3-cycle, then it contains all 3-cycles and hence all of \(A_n\), meaning \(N = A_n\).

Step by step solution

01

Understand the Given Problem

We need to show that if a normal subgroup \(N\) of the alternating group \(A_n\) contains a single 3-cycle (a cycle of the form \((a\ b\ c)\)), then \(N\) must be equal to \(A_n\). Essentially, containing a 3-cycle is sufficient for \(N\) to be entire \(A_n\).
02

Recognize Key Properties of 3-Cycles

Recall that 3-cycles are elements of \(A_n\), and any permutation in \(A_n\) can be expressed as a product of 3-cycles when \(n \geq 3\). Also, 3-cycles generate \(A_n\), meaning that any element of \(A_n\) can be written as a product of 3-cycles.
03

Use Normality of Subgroup

Since \(N\) is a normal subgroup of \(A_n\), any conjugate of a 3-cycle in \(N\) by any element in \(A_n\) will also be in \(N\). That is, if \((a\ b\ c) \in N\), then for any \( \sigma \in A_n\), the conjugate \(\sigma (a\ b\ c) \sigma^{-1} \in N\).
04

Conjugation of 3-Cycles in \(A_n\)

In \(A_n\), conjugating a 3-cycle by other elements can generate any other 3-cycle in \(A_n\). This is because 3-cycles are transitive in \(A_n\) through conjugation, meaning that through appropriate conjugations, we can transform any 3-cycle into any other 3-cycle.
05

Generate All of \(A_n\)

Since \(N\) is closed under conjugation and contains one 3-cycle, through conjugation, \(N\) must contain all possible 3-cycles. So, \(N\) has enough elements to generate \(A_n\), implying \(N = A_n\). This uses the fact that 3-cycles generate \(A_n\).
06

Conclusion: Review the Proof

By establishing that one 3-cycle leads to containing every 3-cycle in \(A_n\), and since these generate \(A_n\), \(N = A_n\). Thus, a single 3-cycle in a normal subgroup forces it to be the entire group.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

3-cycles in group theory
In group theory, a 3-cycle is a type of permutation cycle that involves three elements. For example, the cycle \( (a\ b\ c) \) denotes a permutation where \( a \) goes to \( b \), \( b \) goes to \( c \), and \( c \) goes back to \( a \.\) These are considered basic building blocks in permutation groups.

**Properties of 3-cycles:**
  • A 3-cycle can be represented as a product of transpositions, such as \( (a\ b)(b\ c) \).
  • In groups like the symmetric group \( S_n \,\) 3-cycles play a crucial role in how elements are formed.
  • Importantly, 3-cycles can generate many other elements and are fundamental in the study of alternating groups.
Understanding 3-cycles helps decode the nature of more complex permutations and is essential for grasping the structure of certain mathematical groups.
alternating group
The alternating group, often denoted as \( A_n \,\) is a critical concept in group theory. It consists of the even permutations of a finite set with \( n \) elements. "Even" permutations are those that can be expressed as an even number of transpositions.

**Key Features of the Alternating Group:**
  • For \( n \geq 3 \,\) the group \( A_n \) is non-abelian, meaning its group operations are not commutative.
  • Each alternating group is a subgroup of the symmetric group \( S_n \), comprising half its elements.
  • In particular, \( A_n \) is notable because it's generated by its 3-cycles - any permutation in \( A_n \) can be expressed as a product of 3-cycles.
The alternating group is foundational in fields such as algebra and combinatorics, providing insights into permutations and symmetry.
conjugation in groups
Conjugation is a concept from group theory where an element of the group is transformed via another group element. If \( g \) and \( x \) are elements of a group \( G \,\) the conjugation of \( x \) by \( g \) is given by \( gxg^{-1} \.\) This is akin to transforming or shifting elements in the group.

**Features of Conjugation:**
  • Conjugation is an equivalence relation, meaning it is reflexive, symmetric, and transitive.
  • In a normal subgroup, any element remains even after conjugation by any element of the larger group.
  • For permutations, conjugation maintains the structure but alters the elements involved.
Conjugation helps in analyzing how elements relate within the group and is pivotal in proving whether a subset is normal, such as in our exercise.
generators of a group
A group generator is an element, or a set of elements, from which every other element of the group can be derived through operation combinations. Identifying generators is crucial as they simplify understanding a group's structure.

**Essentials of Group Generators:**
  • When 3-cycles are mentioned as generators of the alternating group \( A_n \,\) it means any element in \( A_n \) can be expressed entirely using products of these 3-cycles.
  • Generators can help express complex permutations simply and effectively.
  • The concept is analogous to a power base, with which you can describe the whole structure with only a few essential elements.
Recognizing the generator allows one to see how all elements relate to each other within the group and provides a powerful way to navigate group relationships.

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Most popular questions from this chapter

Show that a group cannot be written as the set-theoretic union of two proper subgroups.

Let \(G\) be a group, \(T\) an automorphism of \(G, N\) a normal subgroup of \(G\). Prove that \((N) T\) is a normal subgroup of \(G\).

If an abelian group has subgroups of orders \(n\) and \(m\), respectively, then show it has a subgroup whose order is the least common multiple of \(n\) and \(m\).

Find all the automorphisms of \(S_{3}\), the symmetric group of degree 3 .

Let \(G\) be a group, \(H\) a subgroup of \(G, T\) an automorphism of \(G\). Let \((H) T=\\{h T \mid h \in H\\} .\) Prove \((H) T\) is a subgroup of \(G\)
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