Problem 24

Question

In Exercises 19 through 25 find the maximum possible order of an element in the indicated group. $$ A_{6} $$

Step-by-Step Solution

Verified

Answer

The maximum possible order of an element in $ A_6 $ is 5.

1Step 1: Understand the Group

The group in question is the alternating group $ A_6 $, which consists of all even permutations of six elements. In particular, $ A_6 $ is a normal subgroup of the symmetric group $ S_6 $ and has an order of 360, since $ |A_6| = \frac{6!}{2} = 720/2= 360 $.

2Step 2: Determine Orders of Elements

Permutations in $ A_6 $ are composed of 3-cycles and products of disjoint transpositions, each counting as an even permutation. The order of any element in symmetric or alternating groups is the least common multiple of the lengths of the cycles in its disjoint cycle decomposition.

3Step 3: Consider Cycle Structures

For $ A_6 $, possible cycle types include a 5-cycle (e.g., $(1 ext{ } 2 ext{ } 3 ext{ } 4 ext{ } 5)$ or a product of two disjoint 3-cycles (e.g., $(1 ext{ } 2 ext{ } 3)(4 ext{ } 5 ext{ } 6)$). These permutations are both even permutations.

4Step 4: Calculate Orders of Permutations

The order of a 5-cycle is 5, since it returns to identity after 5 applications. The order of a product of two disjoint 3-cycles is 3, since each cycle needs to be completed simultaneously in 3 moves.

5Step 5: Determine Maximum Possible Order

Compare the possible orders: the highest order found from above permutations is from a 5-cycle, which has order 5.

Key Concepts

Alternating GroupsCycle DecompositionPermutationsSymmetric Groups

Alternating Groups

Alternating groups are fascinating structures in group theory, primarily consisting of all even permutations of a given number of elements. Specifically, an alternating group on $ n $ elements is denoted by $ A_n $. For instance, the alternating group $ A_6 $ consists of all even permutations of six elements. This means that each permutation can be expressed as a product of an even number of transpositions.

Alternating groups are normal subgroups of their corresponding symmetric groups.
The order (or size) of $ A_n $ is given by $ |A_n| = \frac{n!}{2} $, where $ n! $ represents the factorial of $ n $.
In $ A_6 $, the order is $ 360 $ because $ |A_6| = \frac{6!}{2} = 720/2 = 360 $.

Alternating groups are particularly important in abstract algebra because they are simple for $ n \geq 5 $, meaning they don’t have normal subgroups other than the trivial group and the group itself.

Cycle Decomposition

Cycle decomposition is a crucial concept when working with permutations. It refers to expressing a permutation as a product of disjoint cycles. Each cycle represents a set of positions that elements are permuted among.

In a disjoint cycle representation, no elements are moved by more than one cycle.
The order of a permutation is the least common multiple (LCM) of the lengths of its disjoint cycles.

Disjoint cycle decompositions help simplify understanding complex permutations by breaking them down into simpler, easily manageable parts. This method reveals the behavior of permutations over iterations and helps in calculating their orders.

Permutations

Permutations are arrangements or reorderings of elements in a set. In mathematics, a permutation of a set is defined as a bijective function from the set to itself.

A permutation is described as even if it can be decomposed into an even number of transpositions (swaps between pairs of elements) and odd if it requires an odd number of such swaps.
Every permutation can be uniquely factored into a product of disjoint cycles.
The maximum order of a permutation in a symmetric or alternating group is determined by the longest cycle in its cycle decomposition.

Understanding permutations is key to studying structures within symmetric and alternating groups since they form the foundation of what these groups entail.

Symmetric Groups

Symmetric groups, denoted by $ S_n $, are essential in group theory and contain all possible permutations of a set with $ n $ elements. Each element in a symmetric group is a permutation, representing one possible rearrangement of the elements.

Symmetric groups have an order equal to $ n! $ (factorial of $ n $), as they encompass every possible permutation of the $ n $ elements.
They include both even and odd permutations, unlike alternating groups, which only contain even permutations.
Symmetric groups are vital in understanding not only abstract algebra but also in solving various problems involving combinatorics and symmetry.

The insights drawn from symmetric groups pave the way for exploring more specific types of groups, such as alternating groups, and play a critical role in higher-level mathematical studies.

Problem 24

Other exercises in this chapter

Problem 23

In the dihedral groups $D_{n}$ with $n \geq 3$, show that we have $\rho \tau=\tau \rho^{-1}$.

View solution

Problem 24

Let $a$ and $b$ be elements of a group $G$ with $|a|=14$ and $|b|=15$. Describe the subgroup $\langle a\rangle \cap\langle b\rangle$. Explain your a

View solution

Problem 24

Let $G=\left\\{a+b i \mid a, b \in \mathbb{R}, a^{2}+b^{2}=1\right\\} .$ Determine whether or not $G$ is a subgroup of $\mathbb{C}^{*}$ under multiplicati

View solution

Problem 24

Prove that a finite group is Abelian if and only if its group table is a symmetric matrix, that is, a matrix $\left\\{a_{i j}\right\\}$ where \(a_{i j}=a_{j i

View solution