Could you explain how to show this(6.16) in detail? ### Problem 6.16**Objective**: Find the orders of all the elements of the alternating group \(A_8\).**Solution Outline**:1. Identify the structure of the alternating group \(A_8\) as the group of even permutations of the symmetric group \(S_8\).2. Determine the cycle types in \(A_8\) and compute their orders.3. Consider permutations as products of disjoint cycles and calculate each element's order as the least common multiple (LCM) of the cycle lengths.**Key Concepts**:- **Even Permutations**: Permutations that can be expressed as an even number of transpositions.- **Cycle Decomposition**: Each permutation can be broken down into disjoint cycles.- **Order of a Permutation**: The smallest positive integer \(m\) such that applying the permutation \(m\) times returns to the identity.**Calculation Examples**:- A single 8-cycle permutation (if existed in \(S_8\), but not in \(A_8\)) has the order 8.- Products such as a 3-cycle and a 5-cycle, or a pair of 4-cycles. Their orders are determined by finding the LCM of cycle lengths due to factorization properties in permutations.Further analysis of specific cycle decompositions in \(A_8\) ensures completeness and accurate computation of all element orders.

Name: Could you explain how to show this(6.16) in detail? ### Problem 6.16**
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: Could you explain how to show this(6.16) in detail? ### Problem 6.16**Objective**: Find the orders of all the elements of the alternating group \(A_8\).**Solution Outline**:1. Identify the structure of the alternating group \(A_8\) as the group of ev