Homomorphisms from S4 to S6

To find the number of homomorphisms from S4 (the symmetric group on 4 letters) to S6 (the symmetric group on 6 letters), we need to consider the properties of group homomorphisms and the structures of the two groups.

Properties of Group Homomorphisms:
- A group homomorphism is a function between two groups that preserves the group structure.
- For a homomorphism φ: G -> H, the following properties hold:
1. φ(eG) = eH, where eG and eH are the identity elements of groups G and H, respectively.
2. φ(g1 * g2) = φ(g1) * φ(g2), where * denotes the group operation in both G and H.

Structure of S4:
- S4 is the group of all possible permutations of 4 elements.
- It has 4! = 24 elements.
- The order of S4 is 24.

Structure of S6:
- S6 is the group of all possible permutations of 6 elements.
- It has 6! = 720 elements.
- The order of S6 is 720.

Counting Homomorphisms:
Since S4 has fewer elements than S6, there cannot be a surjective (onto) homomorphism from S4 to S6. This is because the order of the image of a group under a homomorphism divides the order of the group itself.

To count the number of homomorphisms, we consider the possible images of the generators of S4 under a homomorphism to S6.

Generators of S4:
- S4 is generated by the two elements (12) and (1234), where (12) represents a transposition and (1234) represents a 4-cycle.
- The element (12) has order 2, and the element (1234) has order 4.

Possible Images:
- The identity element e of S4 must be mapped to the identity element e' of S6.
- The element (12) can be mapped to any element of order 2 in S6.
- The element (1234) can be mapped to any element of order 2 or 4 in S6.

Counting:
- The number of choices for the image of (12) is the number of elements of order 2 in S6, which is 6!/(2!2!2!) = 90.
- The number of choices for the image of (1234) is the number of elements of order 2 or 4 in S6, which is 6!/(2!2!) = 180.

Total Number of Homomorphisms:
- By the rule of product, the total number of homomorphisms from S4 to S6 is the product of the choices for the two generators: 90 * 180 = 16,200.

Therefore, there are 16,200 homomorphisms from S4 to S6.