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In group theory, which is a branch of abstract algebra, the concepts of centralizer and normalizer help us understand the structure of groups and their subgroups. Here are the definitions of both: ### Centralizer The centralizer of a subset \( S \) of a group \( G \), denoted as \( C_G(S) \), is the set of all elements in \( G \) that commute with every element of \( S \).