(i) When each is the subject of a sentence the verb is singular and so is any pronoun:
Each has a room to himself.
When a plural noun or pronoun is the subject, with each in apposition, the verb is plural:
They have a room each.
(ii) There is a belief that each other is the right expression when only two persons or things are referred to, and one another when there are more than two. But Fowler, quoted with approval by Jespersen, says of this so-called rule "This differentiation is neither of present utility nor based on historical usage".
For between each, see BETWEEN.
See also THEY FOR HE OR SHE.
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