Mathematics · Group Theory - Subgroups and Conjugacy

Which of the following statements are correct regarding group properties and specific subsets within a group? I. In group theory, the inverse of a product of two elements follows the reversal property, expressed algebraically as (ab)-1 = b-1a-1. II. Group properties dictate that if ab = ac, it implies b = c; however, the relation ba = ca also implies that b = c by cancellation laws. III. The normalizer of an element 'a' is defined as the set of all elements 'g' in a group G such that ag = ga. IV. The conjugate of a subgroup H by an element 'a' is denoted as aHa-1 and is defined as the set {aha-1 : h ∈ H}. Choose the most appropriate answer from the options given below: