Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that for any finite group , the number of elements of every subgroup of divides the number of elements of . Cosets of a particular type of subgroup (a normal subgroup) can be used as the elements of another group called a quotient group or factor group. Cosets also appear in other areas of mathematics such as vector spaces and error-correcting codes.
coSEtS aRE a basIC tool iN ThE STUDY of groUpS; For eXAmPLE, they Play a cEnTral RoLE IN lagrangE'S Theorem tHaT states that For any FINiTE group , the NumBer oF eLeMents of EVERy sUBGRouP of dividES The nUMber of elements of . coSETs of a particulaR tYpE oF SuBGROuP (A norMAL SuBgroup) cAn BE uSEd as tHe elemEnts of anOthER grOup cALlEd A qUoTienT gROup oR factOR grOup. cosEts alsO Appear In other aREas of mathemaTics suCH as vectOr spaces aNd error-coRrECtinG coDes.