Some group (G, o) is known to be abelian. Then, which one of the following is true for G? A. g = g^(-1) for every g in G...

Question

Some group (G, o) is known to be abelian. Then, which one of the following is true for G? A. g = g^(-1) for every g in G  B. g = g^2 for every g in G  C. (g o h)^2 = g^2 o h^2 for every g, h in G  D. G is of finite order

Accepted Answer

C. (g o h)^2 = g^2 o h^2 for every g, h in G A group is abelian when its operation is commutative, i.e. a o b = b o a for all elements. The trick here is to test which statement follows from commutativity alone, without any extra assumption. Options A, B and D each force a special structure (every element is its own inverse, every element is idempotent, or finiteness), but commutativity by itself only lets us reorder factors, which is exactly what turns (g o h)^2 into g^2 o h^2.