quotient group

Before defining quotient groupsMathworldPlanetmath, some preliminary definitions must be introduced and a few established.

Given a group G and a subgroupMathworldPlanetmathPlanetmath H of G, the http://planetmath.org/node/122relationMathworldPlanetmathPlanetmath L on G defined by aLb if and only if b-1aH is called left congruencePlanetmathPlanetmathPlanetmathPlanetmath modulo H; similarly the relation defined by aRb if and only if ab-1H is called congruence modulo H (observe that these two relations coincide if G is abelianMathworldPlanetmath).


Left (resp. right) congruence modulo H is an equivalence relationMathworldPlanetmath on G.


We will only give the proof for left congruence modulo H, as the for right congruence modulo H is analogous. Given aG, because H is a subgroup, H contains the identityPlanetmathPlanetmathPlanetmathPlanetmath e of G, so that a-1a=eH; thus aLa, so L is http://planetmath.org/node/1644reflexiveMathworldPlanetmathPlanetmath. If bG satisfies aLb, so that b-1aH, then by the of H under the formation of inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmath, a-1b=(b-1a)-1H, and bLa; thus L is symmetricMathworldPlanetmathPlanetmath. Finally, if cG, aLb, and bLc, then we have b-1a,c-1bH, and the closure of H under the binary operationMathworldPlanetmath of G gives c-1a=(c-1b)(b-1a)H, so that aLc, from which it follows that L is http://planetmath.org/node/1669transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath, hence an equivalence relation. ∎

It follows from the preceding that G is partitioned into mutually disjoint, non-empty equivalence classesMathworldPlanetmath by left (resp. right) congruence modulo H, where a,bG are in the same equivalence class if and only if aLb (resp. aRb); focusing on left congruence modulo H, if we denote by a¯ the equivalence class containing a under L, we see that

a¯={bGbLa}={bGa-1bH}={bGb=ah for some hH}={ahhH}.

Thus the equivalence class under L containing a is simply the left cosetMathworldPlanetmath aH of H in G. Similarly the equivalence class under R containing a is the right coset Ha of H in G (when the binary operation of G is written additively, our notation for left and right cosets becomes a+H={a+hhH} and H+a={h+ahH}). Observe that the equivalence class under either L or R containing e is eH=H. The index of H in G, denoted by |G:H|, is the cardinality of the set G/H (read “G modulo H” or just “G mod H”) of left cosets of H in G (in fact, one may demonstrate the existence of a bijectionMathworldPlanetmath between the set of left cosets of H in G and the set of right cosets of H in G, so that we may well take |G:H| to be the cardinality of the set of right cosets of H in G).

We now attempt to impose a group on G/H by taking the of the left cosets containing the elements a and b, respectively, to be the left coset containing the element ab; 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 G/H. For the of left coset to be well-defined, we must be sure that if aH=aH and bH=bH, i.e., if aaH and bbH, then abH=abH, i.e., that ababH. Precisely what must be required of the subgroup H to ensure the of the above condition is the content of the following :


The rule (aH,bH)abH gives a well-defined binary operation on G/H if and only if H is a normal subgroupMathworldPlanetmath of G.


Suppose first that of left cosets is well-defined by the given rule, i.e, that given aaH and bbH, we have abH=abH, and let gG and hH. Putting a=1, a=h, and b=b=g-1, our hypothesisMathworldPlanetmathPlanetmath gives hg-1H=eg-1H=g-1H; this implies that hg-1g-1H, hence that hg-1=g-1h for some hH. on the left by g gives ghg-1=hH, and because g and h were chosen arbitrarily, we may conclude that gHg-1H for all gG, from which it follows that HG. Conversely, suppose H is normal in G and let aaH and bbH. There exist h1,h2H such that a=ah1 and b=bh1; now, we have


and because b-1h1bH by assumptionPlanetmathPlanetmath, we see that ab=abh, where h=(b-1hb)h2H by the closure of H under in G. Thus ababH, and because left cosets are either disjoint or equal, we may conclude that abH=abH, so that multiplicationPlanetmathPlanetmath of left cosets is indeed a well-defined binary operation on G/H. ∎

The set G/H, where H is a normal subgroup of G, 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 G), and is called a quotient or factor group (more specifically the quotient of G by H). We conclude with several examples of specific quotient groups.


A standard example of a quotient group is /n, the quotient of the of integers by the cyclic subgroup generated by n+; the order of /n is n, and the distinct left cosets of the group are n,1+n,,(n-1)+n.


Although the group Q8 is not abelian, each of its subgroups its normal, so any will suffice for the formation of quotient groups; the quotient Q8/-1, where -1={1,-1} is the cyclic subgroup of Q8 generated by -1, is of order 4, with elements -1,i-1={i,-i},k-1={k,-k} , and j-1={j,-j}. Since each non-identity element of Q8/-1 is of order 2, it is isomorphicPlanetmathPlanetmathPlanetmath to the Klein 4-group V. Because each of i, j, and k has order 4, the quotient of Q8 by any of these subgroups is necessarily cyclic of order 2.


The center of the dihedral groupMathworldPlanetmath D6 of order 12 (with http://planetmath.org/node/2182presentationMathworldPlanetmathPlanetmath r,sr6=s2=1,r-1s=sr) is r3={1,r3}; the elements of the quotient D6/r3 are r3, rr3={r,r4}, r2r3={r2,r5}, sr3={s,sr3}, srr3={sr,sr4}, and sr2r3={sr2,sr5}; because


D6/r3 is non-abelianMathworldPlanetmath, hence must be isomorphic to S3.

Title quotient group
Canonical name QuotientGroup
Date of creation 2013-03-22 12:04:06
Last modified on 2013-03-22 12:04:06
Owner azdbacks4234 (14155)
Last modified by azdbacks4234 (14155)
Numerical id 35
Author azdbacks4234 (14155)
Entry type Definition
Classification msc 20-00
Synonym factor group
Synonym quotient
Related topic Group
Related topic NormalSubgroup
Related topic Subgroup
Related topic EquivalenceRelation
Related topic Coset
Related topic NaturalProjection
Defines left congruence modulo a subgroup
Defines right congruence modulo a subgroup
Defines index