Question

Let A ⊆ \(\Z\) with 0 ∈ A. For r, s ∈ \(\Z\), define
rA = {ra : a ∈ A},   rA + sA = {ra + sb : a, b ∈ A}.
Which of the following conditions imply that A is a subgroup of the additive group \(\Z\) ?

• A. −2A ⊆ A, A + A = A
• B. A = −A, A + 2A = A
• C. A = −A, A + A = A
• D. 2A ⊆ A, A + A = A

Answer 1

The Correct Option is A, B, C

To determine which conditions imply that \( A \) is a subgroup of the additive group \(\Z\), we need to verify the subgroup criteria for each set of given conditions. The subgroup criteria for an additive subgroup \( A \subseteq \Z \) require that:

1. Closure under addition: If \( a, b \in A \), then \( a + b \in A \).
2. Existence of the additive identity: \( 0 \in A \).
3. Closure under additive inverses: If \( a \in A \), then \( -a \in A \).

Let's evaluate the conditions:

Option 1: \(-2A \subseteq A, A + A = A\)

- The condition \( A + A = A \) satisfies the closure under addition, i.e., for any \( a, b \in A \), \( a+b \in A \).
- The condition \(-2A \subseteq A\) implies that for every \( a \in A \), the element \(-2a \in A\). Considering \( 0 \in A \) (given), we have \(-(-a) = a \in A\). Thus, \( -a \in A \) as well, ensuring closure under additive inverses.

Conclusion: Both conditions are satisfied, so this option implies that \( A \) is a subgroup of \(\Z\).

Option 2: \(A = -A, A + 2A = A\)

- The condition \( A = -A \) denotes closure under additive inverses since for any \( a \in A \), we also have \(-a \in A\).
- The condition \( A + 2A = A \) guarantees closure under addition since for any \( a, b \in A \), \( a + 2b \in A \), indicating that \( a + b \) and \( a \) both belong to \( A \).

Conclusion: Both conditions are satisfied, so this option implies that \( A \) is a subgroup of \(\Z\).

Option 3: \(A = -A, A + A = A\)

- The closure under additive inverses is validated by \( A = -A \).
- The condition \( A + A = A \) confirms closure under addition.

Conclusion: Both conditions are satisfied, so this option implies that \( A \) is a subgroup of \(\Z\).

Option 4: \(2A \subseteq A, A + A = A\)

- The condition \( A + A = A \) ensures closure under addition.
- However, the condition \( 2A \subseteq A \) only indicates that every \( 2a \in A \) for \( a \in A \) but does not necessarily imply \( -a \in A \) for \( a \in A \). Thus, closure under additive inverses isn't fully guaranteed by this condition alone.

Conclusion: The closure under inverses is not guaranteed, so this option does not imply that \( A \) is a subgroup of \(\Z\).

Final Answer: The conditions that imply \( A \) is a subgroup are:
−2A ⊆ A, A + A = A,
A = −A, A + 2A = A,
A = −A, A + A = A.