Question:
Let \(<G,*> \) be a group. Then for all a, b, c \(\in\) G
(A). (a*b)*c \(\in\) G
(B). a*b = b*a
(C). a*(b*c) = (a*b)*c
(D). a*b = a*c implies b = c.
Choose the correct answer from the options given below:
Let \(<G,*> \) be a group. Then for all a, b, c \(\in\) G
(A). (a*b)*c \(\in\) G
(B). a*b = b*a
(C). a*(b*c) = (a*b)*c
(D). a*b = a*c implies b = c.
Choose the correct answer from the options given below:
(A). (a*b)*c \(\in\) G
(B). a*b = b*a
(C). a*(b*c) = (a*b)*c
(D). a*b = a*c implies b = c.
Choose the correct answer from the options given below:
Show Hint
Remember the four group axioms: Closure, Associativity, Identity, and Inverse. Commutativity (a*b = b*a) is an extra property that defines an Abelian group, but it's not required for a general group. The cancellation law is a provable consequence of the main axioms.
Updated On: Sep 24, 2025
- (A), (C) and (D) only.
- (A), (B) and (C) only.
- (A) and (C) only.
- (B) and (C) only.
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept:
A group is a fundamental algebraic structure consisting of a set G and a binary operation * that satisfies four axioms: Closure, Associativity, Identity Element, and Inverse Element. This question tests which of the given statements are necessary properties of any group.
Step 2: Detailed Explanation:
Let's analyze each statement based on the definition of a group:
(A) (a*b)*c \(\in\) G: This property is a direct consequence of the Closure axiom. The closure axiom states that for any a, b \(\in\) G, the result a*b is also in G. If we let d = a*b, then d \(\in\) G. Applying closure again, d*c = (a*b)*c must also be in G. So, statement (A) is correct.
(B) a*b = b*a: This is the commutative property. While some groups have this property (they are called Abelian groups), it is not a requirement for a structure to be a group. For example, the group of invertible matrices under matrix multiplication is not commutative. So, statement (B) is not always true for a general group.
(C) a*(b*c) = (a*b)*c: This is the associative property, which is one of the fundamental axioms of a group. So, statement (C) is correct.
(D) a*b = a*c implies b = c: This is the left cancellation law. It is a property that can be derived from the group axioms and is 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). By associativity, (a\(^{-1}\)*a)*b = (a\(^{-1}\)*a)*c. This simplifies to e*b = e*c, where e is the identity element, so b = c). So, statement (D) is correct.
Step 3: Final Answer:
The properties that hold true for any group are (A), (C), and (D). Statement (B) only holds for Abelian groups. Therefore, the correct option is (1).
A group is a fundamental algebraic structure consisting of a set G and a binary operation * that satisfies four axioms: Closure, Associativity, Identity Element, and Inverse Element. This question tests which of the given statements are necessary properties of any group.
Step 2: Detailed Explanation:
Let's analyze each statement based on the definition of a group:
(A) (a*b)*c \(\in\) G: This property is a direct consequence of the Closure axiom. The closure axiom states that for any a, b \(\in\) G, the result a*b is also in G. If we let d = a*b, then d \(\in\) G. Applying closure again, d*c = (a*b)*c must also be in G. So, statement (A) is correct.
(B) a*b = b*a: This is the commutative property. While some groups have this property (they are called Abelian groups), it is not a requirement for a structure to be a group. For example, the group of invertible matrices under matrix multiplication is not commutative. So, statement (B) is not always true for a general group.
(C) a*(b*c) = (a*b)*c: This is the associative property, which is one of the fundamental axioms of a group. So, statement (C) is correct.
(D) a*b = a*c implies b = c: This is the left cancellation law. It is a property that can be derived from the group axioms and is 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). By associativity, (a\(^{-1}\)*a)*b = (a\(^{-1}\)*a)*c. This simplifies to e*b = e*c, where e is the identity element, so b = c). So, statement (D) is correct.
Step 3: Final Answer:
The properties that hold true for any group are (A), (C), and (D). Statement (B) only holds for Abelian groups. Therefore, the correct option is (1).
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