Question

Compute the numbers \(1 \cdot{ }_{11} 5,2 \cdot{ }_{11} 5,3 \cdot{ }_{11} 5, \ldots, 10 \cdot_{11} 5 .\) Do you get a permutation of the set \(\\{1,2,3,4,5,6,7,8,9,10\\} ?\) Would you get a permutation of the set \(\\{1,2,3,4,5,6,7,8,9,10\\}\) if you used another nonzero member of \(Z_{11}\) in place of \(5 ?\)

Short answer

Yes, you get a permutation. Any nonzero element would work due to properties of finite fields.

Step-by-step solution

Understanding the Task

We need to multiply each of the numbers from 1 to 10 by 5 modulo 11 and check if the results form a permutation of the set \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\). Mathematically, we compute \( n \cdot_{11} 5 = (n \times 5) \mod 11\) for each integer \(n\) from 1 to 10.

Compute Products Modulo 11

Multiply each number from 1 to 10 by 5 and take the result modulo 11:- \(1 \cdot_{11} 5 = (1 \times 5) \mod 11 = 5\)- \(2 \cdot_{11} 5 = (2 \times 5) \mod 11 = 10\)- \(3 \cdot_{11} 5 = (3 \times 5) \mod 11 = 4\)- \(4 \cdot_{11} 5 = (4 \times 5) \mod 11 = 9\)- \(5 \cdot_{11} 5 = (5 \times 5) \mod 11 = 3\)- \(6 \cdot_{11} 5 = (6 \times 5) \mod 11 = 8\)- \(7 \cdot_{11} 5 = (7 \times 5) \mod 11 = 2\)- \(8 \cdot_{11} 5 = (8 \times 5) \mod 11 = 7\)- \(9 \cdot_{11} 5 = (9 \times 5) \mod 11 = 1\)- \(10 \cdot_{11} 5 = (10 \times 5) \mod 11 = 6\)

Check for Permutation

The results of the calculations are \(\{5, 10, 4, 9, 3, 8, 2, 7, 1, 6\}\). Since each number in this list is unique and covers the set \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\) exactly once, these results form a permutation of the set.

Generalizing with Another Element

Since 11 is a prime number, any nonzero element from \(Z_{11}\) will be coprime with 11. By a property of finite fields, multiplication by any such nonzero element will permute the elements of \(Z_{11}\). Hence, any nonzero member can be used in place of 5 to get a permutation.

Key concepts

Permutation

A permutation is essentially a reordering of elements in a set. Imagine having a set of numbers, and then shuffling them to create a new sequence. That new sequence is a permutation. Permutations occur frequently in mathematics when studying sequences and arrangements, as they capture the idea of different arrangements possible for a given set.
In the context of our exercise, we compute the products of numbers from 1 to 10 with 5 (using modular arithmetic), aiming to see if this forms a new arrangement of the same numbers. Since we end up with unique numbers covering the same range, we've achieved a permutation. A fascinating aspect here is how mathematics allows us to explore if these rearrangements still retain the comprehensive nature of the original set.

Finite Fields

Finite fields are a special kind of mathematical structure where only a limited number of elements exist. This limitation means any arithmetic operation on members of the field is computed within a fixed range and 'wraps around' back to the start once reaching the highest number, known as the modulus. Finite fields are frequent in areas like cryptography and coding theory.
In our exercise, we use the finite field denoted as \(Z_{11}\), which consists of elements {0, 1, 2, ..., 10}. The unique feature of a finite field is that every nonzero element has a multiplicative inverse, making it an interesting playground for studying permutations and modular arithmetic.

Multiplicative Group

A multiplicative group is a set equipped with a multiplication operation that satisfies certain properties such as closure, associativity, identity, and invertibility. When this notion is applied to finite fields, the nonzero elements form a multiplicative group. This reveals lots of intriguing properties, especially the potential to form permutations by multiplication.
In our example, if we take any nonzero element in the finite field \(Z_{11}\), such as 5, and multiply it by every element in the group using modulo 11, we create a permutation of the set. This is because each element is distinct and all elements are present exactly once, relying on the special properties of finite fields and their multiplicative structures.

Discrete Mathematics

Discrete mathematics is the mathematical study of discrete structures rather than continuous. This branch of mathematics includes topics such as combinatorics, graph theory, and of course, modular arithmetic. Discrete mathematics is fundamental in computer science and related fields due to its focus on halted, distinct elements.
Our exercise around permutations and finite fields is an excellent example of discrete mathematics in action. The computation of numbers modulo a prime is a perfect instance of counting and arrangement problems, which are central themes in discrete mathematics. Hence, understanding these concepts allows students to grasp better how independent systems of numbers can interact under set rules, leading to insightful outcomes like permutations.