Question

Which of the following statements is/are TRUE for a group \( G \)?

• A. If for all \( x, y \in G \), \( (xy)^2 = x^2 y^2 \), then \( G \) is commutative.
• B. If for all \( x \in G \), \( x^2 = 1 \), then \( G \) is commutative. Here, 1 is the identity element of \( G \).
• C. If the order of \( G \) is 2, then \( G \) is commutative.
• D. If \( G \) is commutative, then a subgroup of \( G \) need not be commutative.

Hint:

In group theory, for a group \( G \) to be commutative, the condition \( (xy)^2 = x^2 y^2 \) must hold for all elements \( x, y \in G \). Also, any group of order 2 is automatically commutative, and groups where every element is its own inverse are also commutative.

Answer 1

The Correct Option is A, B, C

Let's evaluate each statement one by one:
Statement (A): If for all \( x, y \in G \), \( (xy)^2 = x^2 y^2 \), then \( G \) is commutative.
This statement is true. If \( (xy)^2 = x^2 y^2 \) for all \( x, y \in G \), we can expand both sides: \[ (xy)(xy) = x^2 y^2. \] This expands to: \[ x (yx) y = x^2 y^2. \] By associativity and simplifying, we can show that \( xy = yx \), i.e., \( G \) is commutative. Hence, Statement (A) is true.
Statement (B): If for all \( x \in G \), \( x^2 = 1 \), then \( G \) is commutative. Here, 1 is the identity element of \( G \).
This statement is true. If \( x^2 = 1 \) for all elements \( x \in G \), then every element is its own inverse. This property ensures that the group is abelian (commutative) because we can show that \( xy = yx \) for all elements \( x, y \in G \). Hence, Statement (B) is true.
Statement (C): If the order of \( G \) is 2, then \( G \) is commutative.
This statement is true. If the order of \( G \) is 2, then \( G \) must be a group with two elements, and any group of order 2 is always commutative. This is because the only possible group of order 2 is \( \{ e, a \} \), where \( a \) is the only non-identity element, and it must satisfy \( a^2 = e \). This group is trivially commutative. Hence, Statement (C) is true.
Statement (D): If \( G \) is commutative, then a subgroup of \( G \) need not be commutative.
This statement is false. If \( G \) is commutative, then every subgroup of \( G \) is also commutative. In a commutative group, all its subgroups must also be commutative. Hence, Statement (D) is incorrect.
Thus, the correct answer is (A), (B), and (C).