Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

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§ 3.8 Cosets, Normal Subgroups, and Factor Groups

 
Proposition 3.8.1. Let H be a subgroup of the group G, and let a,b be elements of G. Then the following conditions are equivalent:

(1) bH = aH;

(2) bH aH;

(3) b is in aH;

(4) a-1b is in H.

 
A result similar to Proposition 3.8.1 holds for right cosets. Let H be a subgroup of the group G, and let a,b belong to G. Then the following conditions are equivalent:

(1) Ha = Hb;  (2) Ha Hb;   (3) a is in Hb;   (4) ab-1 is in H;
(5) ba-1 is in H;   (6) b is in Ha;   (7) Hb Ha.
The index of H in G could also be defined as the number of right cosets of H in G, since there is a one-to-one correspondence between left cosets and right cosets.

 
Definition 3.8.2. Let H be a subgroup of the group G, and let a be an element of G. The set

aH = { x in G | x = ah   for some   h in H }

is called the left coset of H in G determined by a. Similarly, the right coset of H in G determined by a is the set

Ha = { x in G | x = ha   for some   h in H }.

The number of left cosets of H in G is called the index of H in G, and is denoted by [G:H].

 
Proposition 3.8.3. Let N be a normal subgroup of G, and let a,b,c,d be elements of G.
If aN = cN and bN = dN, then abN = cdN.

 
Theorem 3.8.4. If N is a normal subgroup of G, then the set of left cosets of N forms a group under the coset multiplication given by

(aN)(bN) = abN

for all a,b in G.

 
Definition 3.8.5. If N is a normal subgroup of G, then the group of left cosets of N in G is called the factor group of G determined by N. It will be denoted by G/N.

 
Example 3.8.5. Let N be a normal subgroup of G. If a is any element of G, then the order of aN in G/N is the smallest positive integer n such that an belongs to N.

 
Proposition 3.8.6. Let N be a normal subgroup of G.

(a) The natural projection mapping : G -> G/N defined by (x) = xN, for all x in G, is a homomorphism, and ker() = N.

(b) There is a one-to-one correspondence between subgroups of G/N and subgroups of G that contain N. Under this correspondence, normal subgroups correspond to normal subgroups.

 
Proposition 3.8.7. Let H be a subgroup of the group G. The following conditions are equivalent:

(1) H is a normal subgroup of G;

(2) aH = Ha for all a in G;

(3) for all a,b in G, the coset abH is the set theoretic product (aH)(bH);

(4) for all a,b in G, ab-1 is in H if and only if a-1b is in H.

 
Example 3.8.7. Any subgroup of index 2 is normal.

 
Example 3.8.8. If m is a divisor of n, then Zn / mZn Zm.

 
3.8.8. Theorem [Fundamental Homomorphism Theorem] Let G1, G2 be groups.
If : G1 -> G2 is a group homomorphism with K = ker(), then

G1 / K     (G1).

 
3.8.9. Definition The group G is called a simple group if it has no proper nontrivial normal subgroups.



§ 3.8 Cosets, Normal Subgroups, and Factor Groups: Solved problems

The notion of a factor group is one of the most important concepts in abstract algebra. To construct a factor group, we start with a normal subgroup and the equivalence classes it determines. This construction parallels the construction of Zn from Z. The key idea is to begin thinking of equivalence classes as elements in their own right.

In actually using the Fundamental Homomorphism Theorem, it is important to let the theorem do its job, so that it does as much of the hard work as possible. Quite often we need to show that a factor group G/N that we have constructed is isomorphic to another group G1. The easiest way to do this is to just define a homomorphism µ from G to G1, making sure that N is the kernel of µ. If you prove that µ maps G onto G1, then the Fundamental Theorem does the rest of the work, showing that there exists a well-defined isomorphism between the factor group G/N and G1.

The moral of this story is that if you define a function on G rather than G/N, you ordinarily don't need to worry that it is well-defined. On the other hand, if you define a function on the cosets in G/N, the most convenient way is use a formula defined on representatives of the cosets of N. But then you must be careful to prove that the formula you are using does not depend on the particular choice of a representative. That is, you must prove that your formula actually defines a function. Then you must prove that your function is one-to-one, in addition to proving that it is onto and respects the operations in the two groups. Once again, if your function is defined on cosets, it can be much trickier to prove that it is one-to-one than to simply compute the kernel of a homomorphism defined on G.


27. List the cosets of < 7 > in Z16×. Is the factor group Z16× / < 7 > cyclic?     Solution

28. Let G = Z6 × Z4, let H = { (0,0), (0,2) }, and let K = { (0,0), (3,0) }.
    (a) List all cosets of H; list all cosets of K.
    (b) You may assume that any abelian group of order 12 is isomorphic to either Z12 or Z6 × Z2. Which answer is correct for G/H? For G/K?
    Solution

29. Let the dihedral group Dn be given via generators and relations, with generators a of order n and b of order 2, satisfying ba=a-1 b.
    (a) Show that bai = a-i b for all i with 1 i < n.
    (b) Show that any element of the form ai b has order 2.
    (c) List all left cosets and all right cosets of < b >
    Solution

30. Let G = D6 and let N be the subgroup < a3 > = {e, a3 } of G.
    (a) Show that N is a normal subgroup of G.
    (b) Is G/N abelian?
    Solution

31. Let G be the dihedral group D12, and let N = { e,a3,a6,a9 }.
    (a) Prove that N is a normal subgroup of G, and list all cosets of N.
    (b) You may assume that G/N is isomorphic to either Z6 or S3. Which is correct?
    Solution

32. (a) Let G be a group. For a,b in G we say that b is conjugate to a, written b ~ a, if there exists g in G such that b = gag-1. Show that ~ is an equivalence relation on G. The equivalence classes of ~ are called the conjugacy classes of G.
    (b) Show that a subgroup N of G is normal in G if and only if N is a union of conjugacy classes.
    Solution

33. Find the conjugacy classes of D4.     Solution

34. Let G be a group, and let N and H be subgroups of G such that N is normal in G.
    (a) Prove that HN is a subgroup of G.
    (b) Prove that N is a normal subgroup of HN.
    (c) Prove that if H N = { e }, then HN / N is isomorphic to H.
    Solution


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