Given a group and a subgroup of , the http://planetmath.org/node/122relation on defined by if and only if is called left congruence modulo ; similarly the relation defined by if and only if is called congruence modulo (observe that these two relations coincide if is abelian).
Left (resp. right) congruence modulo is an equivalence relation on .
We will only give the proof for left congruence modulo , as the for right congruence modulo is analogous. Given , because is a subgroup, contains the identity of , so that ; thus , so is http://planetmath.org/node/1644reflexive. If satisfies , so that , then by the of under the formation of inverses, , and ; thus is symmetric. Finally, if , , and , then we have , and the closure of under the binary operation of gives , so that , from which it follows that is http://planetmath.org/node/1669transitive, hence an equivalence relation. ∎
It follows from the preceding that is partitioned into mutually disjoint, non-empty equivalence classes by left (resp. right) congruence modulo , where are in the same equivalence class if and only if (resp. ); focusing on left congruence modulo , if we denote by the equivalence class containing under , we see that
Thus the equivalence class under containing is simply the left coset of in . Similarly the equivalence class under containing is the right coset of in (when the binary operation of is written additively, our notation for left and right cosets becomes and ). Observe that the equivalence class under either or containing is . The index of in , denoted by , is
the cardinality of the set (read “ modulo ” or just “ mod ”) of left cosets of in (in fact, one may demonstrate the existence of a bijection
between the set of left cosets of in and the set of right cosets of in , so that we may well take to be the cardinality of the set of right cosets of in ).
We now attempt to impose a group on by taking the of the left cosets containing the elements and , respectively, to be the left coset containing the element ; however, because this definition requires a choice of left coset representatives, there is no guarantee that it will yield a well-defined binary operation on . For the of left coset to be well-defined, we must be sure that if and , i.e., if and , then , i.e., that . Precisely what must be required of the subgroup to ensure the of the above condition is the content of the following :
The rule gives a well-defined binary operation on if and only if is a normal subgroup of .
Suppose first that of left cosets is well-defined by the given rule, i.e, that given and , we have , and let and . Putting , , and , our hypothesis gives ; this implies that , hence that for some . on the left by gives , and because and were chosen arbitrarily, we may conclude that for all , from which it follows that . Conversely, suppose is normal in and let and . There exist such that and ; now, we have
and because by assumption, we see that , where by the closure of under in . Thus , and because left cosets are either disjoint or equal, we may conclude that , so that multiplication of left cosets is indeed a well-defined binary operation on . ∎
The set , where is a normal subgroup of , is readily seen to form a group under the well-defined binary operation of left coset multiplication (the of each group follows from that of ), and is called a quotient or factor group (more specifically the quotient of by ). We conclude with several examples of specific quotient groups.
A standard example of a quotient group is , the quotient of the of integers by the cyclic subgroup generated by ; the order of is , and the distinct left cosets of the group are .
Although the group is not abelian, each of its subgroups its normal, so any will suffice for the formation of quotient groups; the quotient , where is the cyclic subgroup of generated by , is of order , with elements , and . Since each non-identity element of is of order , it is isomorphic to the Klein -group . Because each of , , and has order , the quotient of by any of these subgroups is necessarily cyclic of order .
|Date of creation||2013-03-22 12:04:06|
|Last modified on||2013-03-22 12:04:06|
|Last modified by||azdbacks4234 (14155)|
|Defines||left congruence modulo a subgroup|
|Defines||right congruence modulo a subgroup|