Final answer:
The possible isomorphism types of an abelian group with 27 elements are Z3 x Z3 x Z3, Z9 x Z3, and Z27.
Step-by-step explanation:
An abelian group with 27 elements can be written as Z27, where Z represents the group of integers and 27 represents the number of elements in the group. The possible isomorphism types of Z27 are determined by the prime factorization of 27, which is 3^3. Each prime factor corresponds to a cyclic subgroup of Z27. Therefore, the possible isomorphism types of an abelian group with 27 elements are: Z3 x Z3 x Z3, Z9 x Z3, and Z27.