# Abstract Algebra: Is S(3) Isomorphic to Z(6)?

• oddiseas
In summary, the question asks whether the symmetric group S(3) and the group of integers modulo six Z(6) are isomorphic. The symmetric group is defined as all the different permutations of a set with six elements, while isomorphism requires a one-to-one and onto map between the two groups. However, it is unclear how to apply the binary operation between these two groups. Also, it is uncertain whether each element of the symmetric group should be considered as an ordering of three distinct points or as a single element. It is worth noting that the symmetric group is not commutative while Z(6) is.
oddiseas

## Homework Statement

Is the symmetric group s(3) isomorphic to Z(6), the group of integers modulo six with addition (mod 6) as its binary operation

## Homework Equations

Basically i know that the symmetric group is all the different permutations of this set and that there are six of them. I also know that to be isomorphic is must be a one to one and onto map. But i can't figure out how to apply a binary operation between these two groups. In addition is each element of the symmetric group considered an ordering of three distinct points? or is each permutation considered as "1" element.?

## The Attempt at a Solution

Note,
$$(1\, 2)(1\,3) = (1\,3\,2) \qquad (1\,3)(1\,2)=(1\,2\,3)$$
so $S_3$ isn't commutative, but $\mathbb{Z}_6$ is.

## 1. What is S(3)?

S(3) refers to the symmetric group of order 3, also known as the permutation group on 3 elements. It consists of all possible permutations of the elements (1, 2, 3).

## 2. What is Z(6)?

Z(6) refers to the cyclic group of order 6, also known as the integers modulo 6. It consists of the elements {0, 1, 2, 3, 4, 5} with addition modulo 6 as the binary operation.

## 3. What does it mean for two groups to be isomorphic?

Two groups are considered isomorphic if there exists a bijection (a one-to-one and onto mapping) between their elements that preserves the group structure. In other words, the two groups have the same algebraic properties and operations, but may have different symbols or names for their elements.

## 4. How can we determine if S(3) and Z(6) are isomorphic?

To determine if two groups are isomorphic, we can use the following criteria: 1) the groups must have the same order, 2) the groups must have the same number of elements of each order, and 3) the groups must have the same structure (e.g. both be abelian or both be non-abelian). In the case of S(3) and Z(6), they both have order 6 and have the same number of elements of each order, but they have different structures (S(3) is non-abelian while Z(6) is abelian). Therefore, S(3) and Z(6) are not isomorphic.

## 5. What are some real-life applications of abstract algebra and isomorphisms?

Abstract algebra and isomorphisms have many applications in fields such as cryptography, coding theory, and physics. For example, isomorphisms can be used to encrypt and decrypt messages in cryptography, and in coding theory, isomorphisms can be used to detect and correct transmission errors in data. In physics, isomorphisms can be used to describe and study symmetries and transformations in physical systems.

• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
970
• Calculus and Beyond Homework Help
Replies
1
Views
928
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
3K
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
4K
• Calculus and Beyond Homework Help
Replies
5
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
635