Overview / review of cosets, factor groups, and homomorphisms

For: Math 421 at Northern Illinois University
From: Beachy/Blair, Abstract Algebra, Second Edition
Covering: Sections 3.7 and 3.8

Cosets and normal subgroups

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 3.8.1. 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 is a subset of 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 be elements of G . Then the following conditions are equivalent:
(1) Ha = Hb;
(2) Ha is a subset of 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 is a subset of 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 3.7.5. A subgroup H of the group G is called a normal subgroup if ghg-1 is in H for all h in H and all g in G.

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, the product 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.

Factor groups

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 aNbN = 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 in G, then the order of aN is the smallest positive integer n such that an is in N.

Group homomorphisms

Definition 3.7.1. Let G and G' be groups, and let f : G -> G' be a function. Then f is said to be a group homomorphism if f(ab) = f(a) f(b) for all a,b in G.

Example 3.7.1. (Exponential functions for groups) Let G be any group, and let a be any element of G. Define f : Z -> G by f(n) = an , for all n in Z. This is a group homomorphism from Z to G.
If G is abelian, with its operation denoted additively, then we define f : Z -> G by f(n) = na.

Example 3.7.2. (Linear transformations) Let V and W be vector spaces. Since any vector space is an abelian group under vector addition, any linear transformation between vector spaces is a group homomorphism.

Proposition 3.7.2. If f : G -> G' is a group homomorphism, then
(a) f maps identity to identity, so f(e) = e;
(b) f preserves inverses, so (f(a))-1 = f(a-1) for all a in G;
(c) f preserves powers, so for any integer n and any a in G, f(an) = (f(a))n;
(d) if a is in G and a has order n, then the order of f(a) in G' is a divisor of n.

Example 3.7.4. (Homomorphisms defined on cyclic groups) Let C be a cyclic group, denoted multiplicatively, with generator a. If f : C -> G is any group homomorphism, and f(a) = g, then the formula f(am) = gm must hold. Since every element of C is of the form am for some integer m, this means that f is completely determined by its value on a .
If C is infinite, then for an element g of any group G , the formula f(am) = gm defines a homomorphism.
If | C | = n and g is any element of G whose order is a divisor of n, then the formula f(am) = gm defines a homomorphism.

Example 3.7.5. (Homomorphisms from Zn to Zk) Any homomorphism f : Zn -> Zk is completely determined by f([1]n), and this must be an element [m]k of Zk whose order is a divisor of n. Then the formula f([x]n) = [mx]k, for all [x]n in Zn, defines a homomorphism. Furthermore, every homomorphism from Zn into Zk must be of this form. The image f(Zn) is the cyclic subgroup generated by [m]k.

Definition 3.7.3 Let f : G -> G' be a group homomorphism. Then
{ x in G | f(x) = e }
is called the kernel of f , and is denoted by ker (f).

Proposition 3.7.4 Let f : G -> G be a group homomorphism, with K = ker (f).
(a) K is a normal subgroup of G.
(b) The homomorphism f is one-to-one if and only if K = {e}.

Proposition 3.7.6 Let f : G -> G' be a group homomorphism.
(a) If H is a subgroup of G, then f(H) is a subgroup of G' .
If f is onto and H is normal in G, then f(H) is normal in G'.
(b) If H' is a subgroup of G', then the inverse image of H' is a subgroup of G.
If H' is a normal in G', then the inverse image of H' is normal in G.

Proposition 3.8.6. Let N be a normal subgroup of G.
(a) The natural projection mapping p : G -> G / N defined by p(x) = xN , for all x in G, is a homomorphism, and ker (p) = 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.

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

Theorem 3.8.8. [Fundamental Homomorphism Theorem] Let G , G' be groups. If f : G -> G' is a group homomorphism with K = ker (f), then G / K is isomorphic to the image f(G).

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