Unit 1 · Discrete Structures and Optimization Paper 2 CS · December 2022 Shift 1

Consider the following statements:
A. A group can have zero elements.
B. A group cannot have any idempotent element except the identity element.
C. An isomorphism from a group ⟨G, ⟩ to ⟨G, ⟩ is called an automorphism.
D. If every element in a group is its own inverse, then the group must be abelian.
Choose the correct answer about the four statements given above.

Pick an option to check yourself, or open the answer below.

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Correct answer: A. B, C and D are correct

Explanation

A group must contain the identity, so it is never empty (A false). If a·a = a, cancelling gives a = e (B true). C is the definition of an automorphism. If x = x⁻¹ for all x, then ab = (ab)⁻¹ = b⁻¹a⁻¹ = ba (D true).

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