Step 1: Core Concept:
A group is a fundamental algebraic structure defined by a set G and a binary operation * that adheres to four key axioms: Closure, Associativity, Identity Element, and Inverse Element. This question assesses which statements are essential properties of any group.
Step 2: Breakdown:
Let's evaluate each statement in light of the group definition:
(A) (a*b)*c \(\in\) G: This directly reflects the Closure axiom. The closure axiom ensures that for any a, b \(\in\) G, a*b is also within G. If we define d = a*b, then d \(\in\) G. Applying closure again, d*c = (a*b)*c must also be in G. Hence, statement (A) is correct.
(B) a*b = b*a: This represents the commutative property. While some groups exhibit this property (Abelian groups), it's not a universal requirement for group membership. For instance, the group of invertible matrices under matrix multiplication is non-commutative. Thus, statement (B) isn't always true for a general group.
(C) a*(b*c) = (a*b)*c: This is the associative property, a foundational axiom of a group. Consequently, statement (C) is correct.
(D) a*b = a*c implies b = c: This describes the left cancellation law. This property can be derived from group axioms and is universally true for all groups. (Proof: Since G is a group, an inverse element \(a^{-1}\) exists. Multiply both sides of a*b = a*c on the left by \(a^{-1}\): \(a^{-1}\)*(a*b) = \(a^{-1}\)*(a*c). Using associativity, (\(a^{-1}\)*a)*b = (\(a^{-1}\)*a)*c. This simplifies to e*b = e*c, where e is the identity element, thus b = c). Hence, statement (D) is correct.
Step 3: Conclusion:
The properties always valid for any group are (A), (C), and (D). Statement (B) only applies to Abelian groups. Therefore, the correct option is (1).