Solution Found!
How does the number (up to isomorphism) of Abelian groups
Chapter 11, Problem 16E(choose chapter or problem)
Problem 16E
How does the number (up to isomorphism) of Abelian groups of order n compare with the number (up to isomorphism) of Abelian groups of order m where
a. n = 32 and m = 52?
b. n = 24 and m = 54?
c. n = pr and m = qr, where p and q are prime?
d. n = pr and m = prq, where p and q are distinct primes?
e. n = pr and m = prq2, where p and q are distinct primes?
Questions & Answers
QUESTION:
Problem 16E
How does the number (up to isomorphism) of Abelian groups of order n compare with the number (up to isomorphism) of Abelian groups of order m where
a. n = 32 and m = 52?
b. n = 24 and m = 54?
c. n = pr and m = qr, where p and q are prime?
d. n = pr and m = prq, where p and q are distinct primes?
e. n = pr and m = prq2, where p and q are distinct primes?
ANSWER:
Step 1 of 5
Part a)
By Fundamental Theorem, there are only two non-isomorphic representatives of Abelian groups of order 
Analogously, there are only two non-isomorphic representatives of Abelian groups of order 
This implies that the number of Abelian groups of these two orders is the same.