When the indefinite pronoun each is the subject, it takes a singular verb.

Each of the research papers is to be written in ten weeks.

When preceded by each, singular subjects joined by and take a singular verb. When followed by each, plural subjects joined by and take a plural verb.

Each cat and dog has been groomed, [singular subjects] The cats and dogs each have been groomed, [plural subjects]

When each is an antecedent, it is referred to with a singular pronoun.

Each of the girls will have to live with her conscience, [not their]

(See also agreement of pronouns and antecedents and agreement of subjects and verbs.)

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