GROUPS

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

Sections 3.7 and 3.8

Cosets and normal subgroups
Factor groups
Group homomorphisms
Some group multiplication tables

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Cosets and normal subgroups

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

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

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

3.8.1. Proposition 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 aH;

(4) a-1b 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,bG. Then the following conditions are equivalent:
(1) Ha = Hb; (2) Ha Hb; (3) a Hb; (4) ab-1 H;
(5) ba-1 H; (6) b 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.

3.7.5. Definition A subgroup H of the group G is called a normal subgroup if

ghg-1 H

for all h H and g G.

3.8.7. Proposition 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 G;

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

(4) for all a,b G, ab-1 H if and only if a-1b H.
Example 3.8.7. Any subgroup of index 2 is normal.

Factor groups

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

3.8.4. Theorem 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 G.

3.8.5. Definition 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 G, then the order of aN in G/N is the smallest positive integer n such that an N.

Group homomorphisms

3.7.1. Definition Let G1 and G2 be groups, and let : G1 -> G2 be a function. Then is said to be a group homomorphism if

(ab) = (a) (b) for all a,b G1.

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

3.7.2. Proposition If : G1 -> G2 is a group homomorphism, then

(a) (e) = e;

(b) ((a))-1 = (a-1) for all a G 1;

(c) for any integer n and any a G1, we have (an) = ((a))n;

(d) if a G1 and a has order n, then the order of (a) in G2 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 : C -> G is any group homomorphism, and (a) = g, then the formula (am) = gm must hold. Since every element of C is of the form am for some integer m, this means that is completely determined by its value on a.
If C is infinite, then for an element g of any group G, the formula (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 (am) = gm defines a homomorphism.

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

3.7.3 Definition Let : G1 -> G2 be a group homomorphism. Then

{ x G1 | (x) = e }

is called the kernel of , and is denoted by ker().

3.7.4 Proposition Let : G1 -> G2 be a group homomorphism, with K = ker().

(a) K is a normal subgroup of G.

(b) The homomorphism is one-to-one if and only if K = {e}.
3.7.6 Proposition Let : G1 -> G2 be a group homomorphism.
(a) If H1 is a subgroup of G1, then (H1) is a subgroup of G2.
If is onto and H1 is normal in G1, then (H1) is normal in G2.

(b) If H2 is a subgroup of G2, then

-1 (H2) = { x G1 | (x) H2 }

is a subgroup of G1.
If H2 is normal in G2, then -1(H2) is normal in G1.
3.8.6. Proposition Let N be a normal subgroup of G.
(a) The natural projection mapping : G -> G/N defined by (x) = xN, for all x 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.
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.

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