2MA105 Algebraic Structures I

Per-Anders Svensson

http://homepage.lnu.se/staff/psvmsi/2MA105.html

Lecture 2

Contents

Groups Definition and Examples

Some Fundamental Properties of Groups

Powers

Subgroups

November ,

Groups Definition and Examples

Definition (Group)Let G a non-empty set and a binary operation on G . The pair(G , ) is called a group, if

is associative,

G contains an identity element with respect to ,

each element in G has an inverse in G , with respect to .

If in addition, is commutative, then (G , ) is called an Abelian (orcommutative) group. The number of element in a group G (if this isfinite) is called the order of G and is written |G |.

Note that the identity element of a group is uniquely determined andthat each element in a group has exactly one inverse.

Example(Z, +), (Q, +), (R, +), (C, +), (Q, ), (R, ), and (C, ) are allexamples of groups. All of them are Abelian.

November ,

ExampleThe set Zn of all residue classes modulo n is an Abelian group underaddition modulo n, but not under multiplication modulo n. Thishowever holds for the so-called group of units modulo n, defined by

Zn = {[a] Zn | gcd(a,n) = 1}.

For example, Z9 = {1, 2, 4, 5, 7, 8} is an Abelian group undermultiplication modulo 9. Its Cayley table is given by

1 2 4 5 7 8

1 1 2 4 5 7 8

2 2 4 8 1 5 7

4 4 8 7 2 1 5

5 5 1 2 7 8 4

7 7 5 1 8 4 2

8 8 7 5 4 2 1

November ,

ExampleLet ABCD be a square in R2. By rotating the square around itsmidpoint or flipping it around a line of symmetry, the square can bemapped onto itself in eight different ways.

A

B C

D D

A B

C C

D A

B B

C D

A

A

D C

B D

C B

A C

B A

D B

A D

C

Using composition of mappings as a binary operation on the set of allthese mappings, we obtain a group; the group of symmetries of asquare, or the dihedral group D4.

Is this group Abelian?

November ,

Some Fundamental Properties of Groups

Theorem (Cancellation Laws)In a group G the cancellation laws

a b = a c = b = c

and

b a = c a = b = c

holds, for all a, b, c G.

Proof.The left cancellation law follows by

a b = a c a1 (a b) = a1 (a c) existence of inverse

(a1 a) b = (a1 a) c is associative

e b = e c

b = c identity element

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TheoremLet G be a group. Then for all a, b G, the equations

a x = b and y a = b

has unique solutions in G (given by x = a1 b and y = b a1,respectively).

Proof.That x = a1 b really is a solution of a x = b is confirmed byplugging this value of x into the equation:

a x = a (a1 b) = (a a1) b = e b = b.

To show uniqueness, assume that there are two solutions x = x1 andx = x2 of the equation. Then a x1 = b and a x2 = b, whencea x1 = a x2. Applying the left cancellation law yields x1 = x2.

November ,

Powers

We will mostly use multiplicative notation, when dealing with groups,which means that we write ab instead of a b. The identity elementof the group G will be denoted 1G .

Sometimes, we will however use additive notation. We then replacea b by a + b, and the identity element of the group G will bewritten 0G . When additive notation is used, it is understood that thegroup in question is Abelian.

Using multiplicative notation we now define powers in the followingway. If n is a positive integer, we put

an = aa . . . a

n factors

Negative powers are defined as

an = a1a1 . . . a1

n factors

= (a1)n .

Moreover, we puta0 = 1G .

November ,

It is now possible to prove the following laws: For all integers mand n we have

an = (an)1, i.e. an is the inverse of an

aman = am+n

(am)n = amn

(ab)n = anbn , if and only if ab = ba.

November ,

Subgroups

ExampleWe take a closer look at the group (Z6, +), and its Cayley table:

+ 0 1 2 3 4 5

0 0 1 2 3 4 5

1 1 2 3 4 5 0

2 2 3 4 5 0 1

3 3 4 5 0 1 2

4 4 5 0 1 2 3

5 5 0 1 2 3 4

If we restrict the binary operation of Z6 only to act on the elements ofthe subset H = {0, 2, 4}, it turns out that H in itself is a group,having the following Cayley table:

+ 0 2 4

0 0 2 4

2 2 4 0

4 4 0 2

We have here an example of a so-called subgroup of Z6.

November ,

Definition (Subgroup)Let (G , ) be a group and H a subset of G . If the restriction of to H defines H as a group, then H is said to be a subgroup of G . Wewrite H G to denote that H is a subgroup of G .

ExampleAs groups under addition we have Z Q R C. As groups undermultiplication we conclude that Q R C.

TheoremLet G be a group and H a subset of G. Then H G, if and only ifthe following three conditions are fulfilled:

(i) H is closed with respect to the binary operation on G, whichmeans that if h1, h2 H , then we also have h1h2 H

(ii) 1G H

(iii) if h H , then h1 H .

November ,

ExampleLet n be a positive integer, and put

nZ = {nk | k Z}.

This is the set of all integer multiples of n. We will show that nZ Z.

Closure We want to show that if h1, h2 nZ, then h1 + h2 nZ.Now there are integers k1, k2 such that h1 = nk1 ochh2 = nk2. Thereforeh1 + h2 = nk1 + nk2 = n(k1 + k2) nZ.

Identity element The identity element of Z is 0, which belongs to nZ,since 0 = n 0.

Inverses Let h nZ. We want to show that the inverse of h, i.e..h, belongs to nZ. But h = nk for some integer k , andthereby h = nk = n (k) nZ.

We have thereby showed that nZ Z.

November ,

ExampleLet G be a group. Put

Z (G) = {z G | zg = gz for all g G}.

We will show that Z (G) G .

Closure Let z1, z2 Z (G). Then z1g = gz1 and z2g = gz2 for allg G . We want to show that z1z2 Z (G), i.e. thatz1z2g = gz1z2 for all g G . This is clear since

z1z2g = z1gz2 = gz1z2.

Identity element The identity element of G belongs to Z (G), since1Gg = g1G = g for all g G .

Inverse Let z Z (G). Then gz = zg for all g G , and we wishto prove that z1g = gz1 for all g G . This followsfrom

gz = zg = z1gzz1 = z1zgz1 = z1g = gz1.

November ,

Groups Definition and ExamplesSome Fundamental Properties of GroupsPowersSubgroups