Problem 7 Find up to isomorphism all Abeli... [FREE SOLUTION]

Chapter 3: Problem 7

Find up to isomorphism all Abelian groups of the indicated orders. $$ n=60 $$

Short Answer

Expert verified

There are 9 non-isomorphic Abelian groups for order 60: $ \mathbb{Z}_{60}, \mathbb{Z}_{30} \times \mathbb{Z}_{2}, \mathbb{Z}_{20} \times \mathbb{Z}_{3}, \mathbb{Z}_{15} \times \mathbb{Z}_{4}, \mathbb{Z}_{12} \times \mathbb{Z}_{5}, \mathbb{Z}_{10} \times \mathbb{Z}_{6}, \mathbb{Z}_{6} \times \mathbb{Z}_{5} \times \mathbb{Z}_{2}, \mathbb{Z}_{4} \times \mathbb{Z}_{3} \times \mathbb{Z}_{5}, \mathbb{Z}_{3} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{5} $.

Step by step solution

Prime Factorization of 60

To find the structure of Abelian groups of a given order, first perform a prime factorization of the integer 60. 60 can be factorized as $ 60 = 2^2 \times 3^1 \times 5^1 $. This shows that 60 is composed of the prime factors 2, 3, and 5.

Using the Fundamental Theorem of Finite Abelian Groups

According to the Fundamental Theorem of Finite Abelian Groups, every finite Abelian group is isomorphic to a direct product of cyclic groups of prime power order.Given the prime factorization of 60, the groups will be of the form $ \igoplus_{i} \ig( \igoplus_{j} \mathbb{Z}_{p^{k_{ij}}} \ig) \igoplus_{k} \big( \igoplus_{l} \mathbb{Z}_{q^{s_{kl}}} \big) \igoplus_{m} \big( \igoplus_{n} \mathbb{Z}_{r^{t_{mn}}} \big) $, where $ p, q, r $ are different primes appearing in the factorization and the summands are cyclic groups.

Determine all Partitions for the Orders of Cyclic Groups

To find all groups, list all partitions of the exponents in the prime factorization:- For $ 2^2 $, partitions of 2: (2), (1+1)- For $ 3^1 $, partition of 1: (1)- For $ 5^1 $, partition of 1: (1)

Construct Groups Using Partitions

Combine the partitions for each prime factor separately and then form groups as direct products:- From partition (2) for $ 2^2 $, we have $ \mathbb{Z}_{4} $. From partition (1+1), we have $ \mathbb{Z}_{2} \times \mathbb{Z}_{2} $.- For $ 3^1 $, we only have $ \mathbb{Z}_{3} $.- For $ 5^1 $, we only have $ \mathbb{Z}_{5} $.Combine all combinations for a total of:1. $ \mathbb{Z}_{60} $2. $ \mathbb{Z}_{30} \times \mathbb{Z}_{2} $3. $ \mathbb{Z}_{20} \times \mathbb{Z}_{3} $4. $ \mathbb{Z}_{15} \times \mathbb{Z}_{4} $5. $ \mathbb{Z}_{12} \times \mathbb{Z}_{5} $6. $ \mathbb{Z}_{10} \times \mathbb{Z}_{6} $7. $ \mathbb{Z}_{6} \times \mathbb{Z}_{5} \times \mathbb{Z}_{2} $8. $ \mathbb{Z}_{4} \times \mathbb{Z}_{3} \times \mathbb{Z}_{5} $9. $ \mathbb{Z}_{3} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{5} $

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Factorization

Prime factorization involves breaking down a number into a product of its prime numbers. For the number 60, the prime factorization is given by \[ 60 = 2^2 \times 3^1 \times 5^1 \]. This means that 60 can be expressed as a product of the primes: 2, 3, and 5.
Prime factorization is foundational because it helps in determining the structure of mathematical groups, particularly Abelian groups. By identifying the prime factors, we know the building blocks and related cyclic groups that can be formed from these primes.
Understanding prime factorization assists with the application of the Fundamental Theorem of Finite Abelian Groups, which we will discuss next.

Finite Abelian Groups

Finite Abelian Groups are a special class of groups distinguished by their commutative nature: the order of operations does not matter. In such groups, \[ a \cdot b = b \cdot a \] for any elements $a$ and $b$ in the group. These groups have a finite number of elements, which makes them particularly interesting when examining direct products of cyclic groups.
The Fundamental Theorem of Finite Abelian Groups states that every finite Abelian group can be expressed as a direct product of cyclic groups whose orders are prime powers. With the prime factorization of 60, we know that Abelian groups of order 60 could be formed by combinations of its prime factors. This theorem provides a structured way to understand the composition and potential group forms possible for Abelian groups of a specified order.

Cyclic Groups

Cyclic groups are groups that can be generated by a single element. This means every element in the group can be expressed as a power of a particular element called the generator.
These groups are fundamental building blocks for other types of groups. They are not only simple but also are easier to work with due to their straightforward structure.
When considering the prime factorization of a number like 60, each factor (e.g., 2, 3, 5) contributes to forming cyclic groups such as $ \mathbb{Z}_4 $, $ \mathbb{Z}_3 $, and $ \mathbb{Z}_5 $. The partitions of the powers of these primes provide various configurations or direct products of these cyclic groups, further elaborating the various Abelian groups possible with a specified order.

Direct Product of Groups

The direct product of groups is a construction used to build more complex groups from simpler ones. If you have multiple groups, you can combine them into a single group where each element of the combined group is a tuple, with each entry from one of the original groups.
In the context of Abelian groups, this concept is essential. For example, given our prime factorization of 60, we can form various group structures like $ \mathbb{Z}_{60}, \mathbb{Z}_{30} \times \mathbb{Z}_{2}, $ and so on. Each of these groups is isomorphic to a direct product of smaller cyclic groups.
The utility of the direct product is that it offers a neat and systematic way to construct new groups, making it a powerful tool in understanding and leveraging the versatility and variability of finite Abelian groups.

91影视

Find up to isomorphism all Abelian groups of the indicated orders. $$ n=60 $$

Short Answer

Step by step solution

Prime Factorization of 60

Using the Fundamental Theorem of Finite Abelian Groups

Determine all Partitions for the Orders of Cyclic Groups

Construct Groups Using Partitions

Key Concepts

Prime Factorization

Finite Abelian Groups

Cyclic Groups

Direct Product of Groups

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