Value of n for One-One Homomorphism from Z12 to Sn

Introduction:
To find the value of n for which there exists a one-one homomorphism from Z12 to Sn, we need to consider the properties of homomorphisms and the structures of Z12 and Sn.

Homomorphisms:
A homomorphism is a map between two algebraic structures that preserves the operations of the structures. In the case of group homomorphisms, it preserves the group operation.

Z12 and Sn:
Z12 is the group of integers modulo 12 under addition. It has 12 elements {0, 1, 2, ..., 11}.
Sn is the symmetric group of order n, which consists of all permutations of n elements.

One-One Homomorphism:
For a homomorphism to be one-one, it must be injective, i.e., distinct elements in the domain map to distinct elements in the codomain.

Finding n:
To find the value of n for which there exists a one-one homomorphism from Z12 to Sn, we need to consider the order of elements in Z12 and Sn.
Since Z12 has 12 elements, we need to find an n such that Sn has at least 12 elements to accommodate a one-one homomorphism.

Conclusion:
In conclusion, the value of n for which there exists a one-one homomorphism from Z12 to Sn is n ≥ 12. This ensures that there are enough elements in Sn to map distinct elements from Z12 in a one-one manner.