quotient group∗
azdbacks4234†
2013-03-21 12:59:28
Before defining quotient groups, some preliminary definitions must be introduced and a few propositions established.
Given a group G and a subgroup H of G, the relation ∼L on G defined by
a ∼L b if and only if b−1 a ∈ H is called left congruence modulo H; similarly the
relation defined by a ∼R b if and only if ab−1 ∈ H is called right congruence
modulo H (observe that these two relations coincide if G is abelian).
Proposition. Left (resp. right) congruence modulo H is an equivalence relation
on G.
Proof. We will only give the proof for left congruence modulo H, as the argument for right congruence modulo H is analogous. Given a ∈ G, because H
is a subgroup, H contains the identity e of G, so that a−1 a = e ∈ H; thus
a ∼L a, so ∼L is reflexive. If b ∈ G satisfies a ∼L b, so that b−1 a ∈ H, then
by the closure of H under the formation of inverses, a−1 b = (b−1 a)−1 ∈ H, and
b ∼L a; thus ∼L is symmetric. Finally, if c ∈ G, a ∼L b, and b ∼L c, then
we have b−1 a, c−1 b ∈ H, and the closure of H under the binary operation of G
gives c−1 a = (c−1 b)(b−1 a) ∈ H, so that a ∼L c, from which it follows that ∼L
is transitive, hence an equivalence relation.
It follows from the preceding proposition that G is partitioned into mutually
disjoint, non-empty equivalence classes by left (resp. right) congruence modulo
H, where a, b ∈ G are in the same equivalence class if and only if a ∼L b
(resp. a ∼R b); focusing on left congruence modulo H, if we denote by ā the
equivalence class containing a under ∼L , we see that
ā = {b ∈ G | b ∼L a}
= {b ∈ G | a−1 b ∈ H}
= {b ∈ G | b = ah for some h ∈ H} = {ah | h ∈ H}.
Thus the equivalence class under ∼L containing a is simply the left coset
aH of H in G. Similarly the equivalence class under ∼R containing a is the
∗ hQuotientGroupi
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setting: h1i hDefinitioni h20-00i
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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 + h | h ∈ H} and
H + a = {h + a | h ∈ H}). 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 bijection
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 product 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 operation of left coset multiplication to be
well-defined, we must be sure that if a0 H = aH and b0 H = bH, i.e., if a0 ∈ aH
and b0 ∈ bH, then a0 b0 H = abH, i.e., that a0 b0 ∈ abH. Precisely what must be
required of the subgroup H to ensure the satisfaction of the above condition is
the content of the following proposition:
Proposition. The rule (aH, bH) 7→ abH gives a well-defined binary operation
on G/H if and only if H is a normal subgroup of G.
Proof. Suppose first that multiplication of left cosets is well-defined by the given
rule, i.e, that given a0 ∈ aH and b0 ∈ bH, we have a0 b0 H = abH, and let g ∈ G
and h ∈ H. Putting a = 1, a0 = h, and b = b0 = g −1 , our hypothesis gives
hg −1 H = eg −1 H = g −1 H; this implies that hg −1 ∈ g −1 H, hence that hg −1 =
g −1 h0 for some h0 ∈ H. Multiplication on the left by g gives ghg −1 = h0 ∈ H,
and because g and h were chosen arbitrarily, we may conclude that gHg −1 ⊆ H
for all g ∈ G, from which it follows that H EG. Conversely, suppose H is normal
in G and let a0 ∈ aH and b0 ∈ bH. There exist h1 , h2 ∈ H such that a0 = ah1
and b0 = bh1 ; now, we have
a0 b0 = ah1 bh2 = a(bb−1 )h1 bh2 = ab(b−1 h1 b)h2 ,
and because b−1 h1 b ∈ H by assumption, we see that a0 b0 = abh, where h =
(b−1 hb)h2 ∈ H by the closure of H under multiplication in G. Thus a0 b0 ∈
abH, and because left cosets are either disjoint or equal, we may conclude that
a0 b0 H = abH, so that multiplication 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
satisfaction of each group axiom 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.
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Example. A standard example of a quotient group is Z/nZ, the quotient of
the additive group of integers by the cyclic subgroup generated by n ∈ Z+ ;
the order of Z/nZ is n, and the distinct left cosets of the group are nZ, 1 +
nZ, . . . , (n − 1) + nZ.
Example. 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 /h−1i, where h−1i = {1, −1} is the cyclic subgroup of Q8 generated by
−1, is of order 4, with elements h−1i, ih−1i = {i, −i}, kh−1i = {k, −k} , and
jh−1i = {j, −j}. Since each non-identity element of Q8 /h−1i is of order 2, it is
isomorphic to the Klein 4-group V . Because each of hii, hji, and hki has order
4, the quotient of Q8 by any of these subgroups is necessarily cyclic of order 2.
Example. The center of the dihedral group D6 of order 12 (with presentation
hr, s | r6 = s2 = 1, r−1 s = sri) is hr3 i = {1, r3 }; the elements of the quotient
D6 /hr3 i are hr3 i, rhr3 i = {r, r4 }, r2 hr3 i = {r2 , r5 }, shr3 i = {s, sr3 }, srhr3 i =
{sr, sr4 }, and sr2 hr3 i = {sr2 , sr5 }; because
sr2 hr3 irhr3 i = sr3 hr3 i = shr3 i =
6 srhr3 i = rhr3 isr2 hr3 i,
D6 /hr3 i is non-abelian, hence must be isomorphic to S3 .
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