each the other; one another (used as a compound reciprocal pronoun): to strike at each other; to hold each other's hands; to love each other.
Origin of each other
before 1000;Middle English; Old English. See each, other
Although some insist that each other be used only in reference to two ( The two candidates respected each other ) and one another in reference to three or more ( The three nations threaten one another ), in standard practice they are interchangeable. Each other is not restricted to two, nor is one another restricted to three or more. The possessive of each other is each other's; the possessive of one another is one another's.
Also, one another. Each one the other, one the other, as in The boys like each other, or The birds were fighting one another over the crumbs. Both of these phrases indicate a reciprocal relationship or action between the subjects preceding (the boys, the birds). Formerly, many authorities held that each other should be confined to a relationship between two subjects only and one another used when there are more than two. Today most do not subscribe to this distinction, which was never strictly observed anyway. [Late 1300s] Also see at each other's throats.