How many homomorphisms between s3 amd s4?

Question

Answer

Homomorphisms between S3 and S4

Let's first define what the symmetric groups S3 and S4 are:

S3: The symmetric group on 3 elements, also known as the permutation group of degree 3, consists of all possible permutations of the three elements {1, 2, 3}. It has 6 elements: {(1), (12), (13), (23), (123), (132)}.

S4: The symmetric group on 4 elements, also known as the permutation group of degree 4, consists of all possible permutations of the four elements {1, 2, 3, 4}. It has 24 elements: {(1), (12), (13), (14), (23), (24), (34), (123), (124), (134), (234), (132), (142), (143), (243), (12)(34), (13)(24), (14)(23), (1234), (1243), (1342), (2341), (1324), (1423)}.

Homomorphisms Definition

A homomorphism is a function between two algebraic structures that preserves the operations of the structures. In the context of symmetric groups, a homomorphism between S3 and S4 is a function that maps elements of S3 to elements of S4 in such a way that the group operations (composition of permutations) are preserved.

Number of Homomorphisms

Let's consider the possible homomorphisms from S3 to S4:

Trivial Homomorphism: The function that maps every element of S3 to the identity element (1) of S4 is a homomorphism. This is the only trivial homomorphism that exists.

Non-Trivial Homomorphisms: Any non-trivial homomorphism from S3 to S4 must map the elements of S3 to permutations in S4 that have the same cycle structure. Since S3 only has elements of cycle lengths 1, 2, and 3, the non-trivial homomorphisms must map to permutations in S4 with cycle lengths 1, 2, and 3 as well.

Since S4 has permutations with cycle lengths 1, 2, 3, and 4, we can choose permutations of cycle lengths 1, 2, and 3 to map the elements of S3. For each element in S3, we have 4 choices in S4. Therefore, for each element in S3, there are 4 possible mappings in S4.

Since S3 has 6 elements, the total number of possible homomorphisms from S3 to S4 is 6 * 4 = 24.

Conclusion

In conclusion, there are a total of 24 possible homomorphisms from S3 to S4. This includes the trivial homomorphism and the non-trivial homomorphisms that preserve the cycle structure of the permutations between the two

Top Courses for Mathematics

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams