Introduction:
To determine whether there exists any onto homomorphism from S5 to S4, we need to analyze the properties of these two symmetric groups. S5 represents the group of all permutations of a 5-element set, while S4 represents the group of all permutations of a 4-element set.

Definition of a Homomorphism:
A homomorphism is a map between two groups that preserves the group operation. In this case, the group operation is composition of permutations. A homomorphism φ from S5 to S4 satisfies the following properties:
1. φ(xy) = φ(x)φ(y) for all x, y in S5.
2. φ(e) = e', where e is the identity element in S5 and e' is the identity element in S4.

Size of the Groups:
The size of S5 is 5! = 120, and the size of S4 is 4! = 24. Since the sizes of the groups are different, an onto homomorphism cannot exist from S5 to S4. This is because an onto homomorphism must map every element of the domain group to a unique element in the codomain group, and the sizes of the groups must be the same for this to be possible.

Pigeonhole Principle:
The Pigeonhole Principle states that if n items are distributed into m containers, and n > m, then at least one container must contain more than one item. In this case, there are 120 elements in S5 and only 24 elements in S4. Therefore, there must be at least one element in S5 that is mapped to more than one element in S4. This violates the condition of an onto homomorphism.

Conclusion:
Based on the size of the groups and the Pigeonhole Principle, we can conclude that there does not exist any onto homomorphism from S5 to S4. The sizes of the groups are different, and the Pigeonhole Principle guarantees that there will be elements in S5 that cannot be uniquely mapped to elements in S4.