Using the noun each properly can be a challenge. Because it is often followed by a prepositional phrase referring to a plural noun, the temptation is to use a plural verb with it. “Each of the team members has his own helmet.” There is a temptation to use the plural havebecause it follows “members”; but if you remove the prepositional phrase “of the team members,” and return to the subject of the sentence, it makes no sense to say “each have.”
A bit more difficult is which pronoun to use in this sentence: “Each of the children went outdoors with (his, her, or their?) mittens on.” If the children were all boys (his) or all girls (her), it would be no in problem; but if it is a mixed group of boys and girls, there is a very strong temptation to use their, even though the reference is to the singular “each.” It vexes English teachers to hear people use their in that circumstance, but the alternatives are not exactly palatable; years back the pronoun his served as a reference to both sexes, but the feminist movement has made that archaic. To use “his or her” is correct but cumbersome, and “his/her” is generally disdained by purists.
It may grate on the ears of traditionalists, but the use of their in reference to a singular noun or pronoun has become so common, if not yet universal, that there are grammarians who have yielded, throwing up their arms, and saying “So what? Their works; use it.”
Although it is not uncommon to see the possessive form of each other’s (or one another’s) followed by a plural noun (“John and Mary admired each others’ cars”), it is incorrect; and it should be “John and Mary admired each other’s car.” Note also it is a common error to place the apostrophe after the “s”(“others’ ” should be the singular “other’s”).
And, before leaving the subject, there is an accepted exception to the rule. Nouns are in “apposition” to each other when they refer to the same subject and are generally side-by-side in a sentence. Where each is in apposition to a plural noun, it takes a plural verb and plural pronoun. “The four file cabinets each have their own colors.” Both the verb have and the pronoun their are properly used. The use of each in that sentence may be unnecessary, strictly speaking, to its meaning, but it adds texture to place it next to “file.” Texture is good.
– Ken Butera