Question

Which of the following subsets form subgroups of the group <ℤ, +>?

(A). H₁ = {0}
(B). H₂ = {n+1 : n ∈ ℤ}
(C). H₃ = {2n : n ∈ ℤ}
(D). H₄ = {2n+1 : n ∈ ℤ}

Choose the correct answer from the options given below:

• A. (A) and (C) only.
• B. (A), (B) and (C) only.
• C. (A), (B), (C) and (D).
• D. (B), (C) and (D) only.

Hint:

The fastest way to check if a subset of \((\mathbb{Z}, +)\) is a subgroup is to see if it can be written in the form \(k\mathbb{Z}\) for some integer \(k\). The set of odd numbers cannot be written in this form and is the classic example of a subset that is not a subgroup because it lacks the identity element (0).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For a non-empty subset H to be a subgroup of the group <ℤ, +>, it must be closed under addition and contain inverses for each of its elements. The identity element in <ℤ, +> is 0, and the inverse of an element a is -a. A key theorem states that all subgroups of <ℤ, +> are of the form kℤ = {kn : n ∈ ℤ} for some non-negative integer k.
Step 2: Detailed Explanation:
We will test each subset against the subgroup criteria:

(A) H₁ = {0}: This is the trivial subgroup. It corresponds to the form kℤ with k=0. It contains the identity (0), is closed under addition (0+0=0), and contains the inverse of 0 (which is 0). Thus, (A) is a subgroup.

(B) H₂ = {n+1 | n ∈ ℤ}: This is another way of writing the set of all integers, ℤ. For any integer k, we can choose n = k-1 (which is an integer), so that n+1 = (k-1)+1 = k. Thus, H₂ = ℤ. A group is always a subgroup of itself (the improper subgroup). This corresponds to kℤ with k=1. Mathematically, this is a subgroup. However, exam questions often distinguish between proper subgroups and the group itself.

(C) H₃ = {2n | n ∈ ℤ}: This is the set of all even integers. This corresponds to the form kℤ with k=2. It contains the identity (0, for n=0), is closed (even + even = even), and contains inverses (the inverse of an even number is also even). Thus, (C) is a subgroup.

(D) H₄ = {2n+1 | n ∈ ℤ}: This is the set of all odd integers. This set does not contain the identity element 0, because 2n+1=0 implies n = -1/2, which is not an integer. A subset that does not contain the identity cannot be a subgroup. Thus, (D) is not a subgroup.

Step 3: Final Answer:
The subsets that are subgroups are H₁, H₂, and H₃. The subset H₄ is not a subgroup.

However, multiple-choice exams often exclude the whole group ℤ itself from “subgroup” options, treating only the trivial subgroup and proper subgroups as valid. Under that convention, the correct answer is: (A) and (C) only.