> )D
!"#$%&'(C,-./0123456789:;<=>?@ABRoot EntrydO)PLwR*PowerPoint Document(TMSummaryInformation(+.DocumentSummaryInformation8p(
4/00DTimes New RomanTT ܖ0ܖDSymbolew RomanTT ܖ0ܖ DChevaraOutlineTT ܖ0ܖ0DWingdingslineTT ܖ0ܖ@.
@n?" dd@ @@``x4NJ*
0AA@g4KdKd 0vppp@<4BdBd x0Tʚ;Sk8ʚ;<4!d!d x0 <4dddd x0 0___PPT10
___PPT9/06 78:$?%;6Examples of Groups $z(Z, +) is an abelian group.
(Q, +) is an abelian group.
(R, +) is an abelian group.
(C, +) is an abelian group.
(Q - {0}, ) is an abelian group.
(Q+, ) is an abelian group.
(R - {0}, ) is an abelian group.
(R+, ) is an abelian group.
(C, - {0}, ) is an abelian group.
Remark: All of the above are groups which are familiar from before in all these groups, the underlying set is infinite. Such groups are known as infinite groups. 9" 9 " #" " " "
<7
Finite Groups$~If (G, *) is a group, and the underlying set G is finite, then we call it a finite group. For a finite group (G, *), the number of elements in G is called the order of the group, written |G|.
Some examples of finite groups:
Let Z6 = {0,1,2,3,4,5} and define a b = (a + b) mod 6 for all a,b Z6. This operation is known as addition mod 6 (modular addition). Then (Z6 , ) is an abelian group.
We can generalize the above example to any positive integer n. Let Zn = {0,1,& .,n 1} and define a b = (a + b) mod n for all a,b Zn. Then (Zn , ) is an abelian group. <" _" =!G] $$$$!$$((((,,000000444444,
=8%Examples of Finite Groups - continued&&$Let K4 = {e,a,b,c} and let * be an operation on K4 defined by the following table (such a table is known as a group composition table). Then it can be verified that all the group axioms are satisfied by (K4 , *), known as Klein s four group.n" +%>:Examples of Groups - Continued$Let Zn* = set of positive integers < n and relatively prime to n, for n 2, that is Zn* = {j: 1 j < n, gcd (j,n) = 1}.
Define the operation on Zn* by a b = a b (mod n) for all a,b Zn* (multiplication modulo n).
Then (Zn* , ) is a group.
Note that (Zn* , ) is a finite group and | Zn* | = j (n), where j is Euler s j function, aka totient function. J{" " O$)%!!D>j 0` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@,|?" dd@ " @ ` n?" dd@ @@``PR @ ` `p>>f(
6Tf P
T Click to edit Master title style!
!
0@i
RClick to edit Master text styles
Second level
Third level
Fourth level
Fifth level!
S
0p ``
>*
08u `
@*
0t `
@*H
0h ? ̙33 Default Design8(
04! P
>*
0$
@*
6l( `P
>*
6, `
@*H
0h ? ̙3380___PPT10.pO&2
0r(
x
c$}"
"
c$<"<$
0"
"p`PpH
0h ? ̙332
0r(
x
c$5"
"
c$3"<$
0"
"p`PpH
0h ? ̙33
0bZ))(
x
c$"
"
c$ : <$
0"
"p`Ppxz @
##"&RLKLK@
<|: ?i
Ie @`
<4/ ?i
Ia @`
<H( ? i
Ib @`
<A ?i
Ic @`
< ?i
Ic @`
<q ?i
Ia @`
<