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: Subgroup Criteria:
For a subset H of the group <ℤ, +> to be a subgroup, it must satisfy two conditions: closure under addition and the inclusion of inverses for all its elements. The identity element in <ℤ, +> is 0, and the inverse of an element *a* is -*a*. All subgroups of <ℤ, +> take the form *k*ℤ = {*kn* : *n* ∈ ℤ} for some non-negative integer *k*.
Step 2: Subset Analysis:
We examine each subset against the subgroup criteria:

(A) H₁ = {0}: This is the trivial subgroup, corresponding to *k* = 0. It contains the identity (0), is closed under addition (0+0=0), and includes the inverse of 0 (which is 0). Hence, (A) is a subgroup.

(B) H₂ = {n+1 | n ∈ ℤ}: This set is equivalent to the set of all integers, ℤ. For any integer *k*, we can find an *n* = *k*-1 (which is also an integer) such that *n*+1 = (*k*-1)+1 = *k*. Thus, H₂ = ℤ. A group is always a subgroup of itself. This corresponds to *k*ℤ with *k*=1. This is mathematically a subgroup. However, exams often differentiate between proper subgroups and the group itself.

(C) H₃ = {2n | n ∈ ℤ}: This is the set of even integers, corresponding to *k* = 2. It contains the identity (0, when *n*=0), is closed under addition (even + even = even), and includes inverses (the inverse of an even number is also even). Therefore, (C) is a subgroup.

(D) H₄ = {2n+1 | n ∈ ℤ}: This represents the set of odd integers. This set lacks the identity element 0, because 2*n*+1=0 implies *n* = -1/2, which is not an integer. A subset without the identity cannot be a subgroup. Consequently, (D) is not a subgroup.

Step 3: Conclusion:
Subgroups are H₁, H₂, and H₃. H₄ is not a subgroup.

However, multiple-choice exams frequently exclude the entire group ℤ from "subgroup" options, considering only the trivial subgroup and proper subgroups valid. In such cases, the correct answer is: (A) and (C) only.