Which of the following statements is/are TRUE for a group G ?
(A)
If for all x, y ∈ G, (xy)2 = x2 y2 then G is commutative.
(B)
If for all x ∈ G, x2= 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
Answer: (A) (B) (C)
Explanation:
A. Given that, (xy)2 = x2y2
xyxy =xxyy
yx = xy (∵ By applying cancelation laws in group)
∴ ∀x, y ∈ G, yx =xy
∴ G is commutative.
B. ∀x ∈ G, x2 = 1
⇒ x = x-1 (∵ x2 = 1, xx =1, x-1xx =x-1, ex =x-1, x=x-1)
In a group if every element has its own inverse then group is commutative.
C. Every group of prime order is commutative so of O(G) =2, the Group is ‘G’ is commutative.
D. If G is commutative then a subgroup of ‘G’ is also commutative.
Let H is a subgroup of group commutative group ‘G’
∀a, b ∈ H, we have a, b, ∈ G and ab = ba(∵ ‘G’ is commutative)
∴ H is commutative.
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