Answer :
Final answer:
The proof entails factorizing the order of the group that gives us two partitions. Each partition corresponds to a distinct abelian group, so there are two abelian groups of order 147.
Explanation:
The question asks to prove that there are two abelian groups of order 147. To do this, we need to use mathematical models in group theory. We have that the order, 147, factors as 3 × 72. The number of abelian groups of this order is given by the number of partitions of the exponent of each prime factor in the prime factorization of the order.
So, considering 72, the partitions are (2), and (1,1). Each corresponds to a distinct abelian group. The (2) partition corresponds to the group Z_72, and the (1,1) partition corresponds to the group Z_7 × Z_7. Therefore, indeed there are two abelian groups of order 147, thereby completing the proof.
Learn more about Abelian Groups here:
https://brainly.com/question/34474352
#SPJ11
