surver of all ten-element groups



           Let G be any group of order 10 Reason as in prceding Exercise to show that G = {e,a,b,b2,b3,b4,ab,ab2,ab3,ab4}, where a has order 2 and b has order 5 


1.) Prove that ab cannot be equal to  e, a, b, b2, b3or b4


Proof 


Case 1  ถ้า      ba = e
		        ba·a = ea
		          ba2 = a
		           be = a
 			 b = a	(ซึ่งเกิดการขัดแย้ง)
	Case 2  ถ้า      ba = a
		        ba·a = a·a
		          ba2 = a2
		           be = e
 			 b = e	(ซึ่งเกิดการขัดแย้ง)
	Case 3  ถ้า       ba = b
		        b-1 ba = b-1b
		              a = e (ซึ่งเกิดการขัดแย้ง)
	Case 4  ถ้า       ba = b2
		       	  a = b (ซึ่งเกิดการขัดแย้ง)
	Case 5  ถ้า       ba = b3
		        b-1 ba = b-1b3
		              a = b2 (ซึ่งเกิดการขัดแย้ง)
	Case 6  ถ้า       ba = b4


		       	  a = b3 (ซึ่งเกิดการขัดแย้ง)
ดังนั้น  	จึงสรุปได้ว่า ba ไม่เท่ากับ e, a, b, b2, b3or b4
2.) Prove that if ba = ab, then G =/ Z10
Proof 

ให้ f : G -------> Z10
กำหนดโดย            	f(e) = [0]                                              	f(b4) = [8]
		f(a) = [5]			f(ab) = [7]
		f(b) = [2]			f(ab2) = [9]
	 	f(b2) = [4]			f(ab3) = [1]
		f(b3) = [6]			f(ab4) = [3]

สามารถเขียนตารางเปรียบเทียบได้ ดังนี้




(G,·)
e
a
b
b2
b3
b4	
ab
ab2
ab3
ab4


e	
e
a	
b	
b2	
b3
b4
ab
ab2
ab3
ab4


a	
a	
e	
Ab
ab2
ab3
ab4
b	
b2
b3
b4


b	
b	
ab
b2
b3
b4
e	
ab2
ab3
ab4
a


b2
b2
ab2
b3
b4
e	
b
ab3
ab4
a
ab


b3
b3
ab3
b4
e	
b	
b2
ab4
a
ab
ab2


b4
b4
ab4
e
b
b2
	b3	
a	
ab
ab2
ab3


ab	
ab
b
Ab2
ab3
ab4
a
	b2
b3
b4
e


ab2
ab2
b2
	Ab3	
ab4
a
ab
b3
b4
e
b


ab3
ab3
b3
Ab4	
a
ab
ab2
b4	
e
b	
b2


ab4
ab4	
b4	
a	
ab
ab2
ab3
e	
b
b2
b3






(Z,·,+)
[0]	
[5]
[2]
[4]	
[6]
[8]
[7]
[9]
[1]
[3]


[0]
[0]	
[5]	
[2]
[4]
[6]	
[8]
[7]	
[9]
[1]	
[3]


[5]
[5]
[0]
[7]
[9]
[1]
[3]
[2]
[4]
[6]
[8]


[2]
[2]
[7]
[4]
[6]
[8]	
[0]
[9]
[1]
[3]
[5]


[4]
[4]
[9]
[6]
[8]
[0]
[2]
[1]	
[3]
[5]
[7]


[6]
[6]
[1]
[8]	
[0]	
[2]
[4]	
[3]
[5]
[7]
[9]


 [8]
[8]
[3]
[0]
	[2]	
[4]
[6]
[5]
[7]
[9]
[1]


[7]
[7]
[2]
[9]
[1]
[3]
[5]
[4]
[6]
[8]
[0]


[9]
[9]
[4]
[1]
	[3]	
[5]
[7]
[6]
[8]
[0]
[2]


[1]	
[1]
[6]
[3]	
[5]
[7]
[9]
[8]	
[0]
[2]
[4]


[3]
[3]
[8]
[5]	
[7]
[9]	
[1]
[0]
[2]	
[4]
[6]





จากตารางจะเห็นได้ว่า G=/ Z10




3.) If ba = ab2 , Prove that ba2= a2b4 and conclude that b =b4 
 This is impossible because b has order 5 , hence  ba  =/ ab2
Proof 
       ให้	ba = ab2
       ต้องพิสูจน์ให้ได้ว่า 	ba2 =/  a2b4
        	    จาก   ba2 = (ba)a
                                         = (ab2)a
                                         = ab(ba)
                                         = a(ba) b2
                                         = aab2b2
                                         = a 2b
                      ดังนั้น   ba2 = a2b4
                      เนื่องจาก  a = 2   จะได้ว่า
                                ba2 = a2b4 
                                   be = eb4
                                     b = b4
4.) If ba = ab3 , Prove that ba2 = a2 b9 = a2 b4 and conclude that b =b4
This is impossible (Why?), hence ba ไม่เท่ากับ ab2
 Proof 

	ให้   ba = ab3
	 ต้องพิสูจน์ให้ได้ว่า  ba2 = a2b9 = a2b4
                                           จาก   ba2 = (ba)a
                                                            = (ab3)a
                                                            = ab2(ba)
                                                            = ab 2ab3
                                                            = ab(ba)b3
                                                            = ab(ab3)b3
                                                            = a(ba)b3b3
                                                            = aa b3b3b3
                                                            = a2b9
                                        ดังนั้น  ba2 = a2b4
                                       เนื่องจาก   b  = 5  จะได้ว่า
                                                   ba2 = a2b4b5
                                           ba2 = a2b9 = a2b4
5.) Prove that if ba=ab4 then G ~= D5 ( where is the group symmetrics of the pentagon)
Proof 

                                                               ให้  f : G ------> D5



 กำหนดโดย f(e)= 

	1
2
3
4
5


1
2
3
4
5

	



 กำหนดโดย f(a)= 

	1
2
3
4
5


2
1
5
4
3

	



 กำหนดโดย f(b)= 

	1
2
3
4
5


2
3
4
5
1

	




 กำหนดโดย f(b2)= 

	1
2
3
4
5


3
4
5
1
2

	



 กำหนดโดย f(b3)= 

	1
2
3
4
5


4
5
1
2
3

	



 กำหนดโดย f(b4)= 

	1
2
3
4
5


4
5
1
2
3

	



	
 กำหนดโดย f(ab)= 

	1
2
3
4
5


3
2
1
5
4

	



 กำหนดโดย f(ab2)= 

	1
2
3
4
5


4
3
2
1
5

	



 กำหนดโดย f(ab3)= 

	1
2
3
4
5


5
4
3
2
1

	



 กำหนดโดย f(ab4)= 

	1
2
3
4
5


1
5
4
3
2

	








D5,· 






























































































































































(G, *)
e
A
b	
b2
b3
b4
ab	
ab2
ab3
ab4


e
e
A
b
b2
b3
b4
ab
ab2
ab3
ab4


a
a
E
ab
ab2
ab3
ab4
b	
b2
b3
	b4


b
b
Ab4
b2
b3
b4	
e	
a	
ab
ab2
ab3


b2
b2
Ab3
b3
b4	
e	
b
ab4
a	
ab
ab2


b3
b3
Ab2
b4	
e	
b	
b2
ab3
ab4
a
	ab


b4
b4
ab	
e
b	
b2	
b3
ab2
ab3
ab4
a


ab
ab
b4	
ab2
ab3
ab4
a
e
b
b2
b3


ab2
ab2
b3
ab3
ab4
a	
ab
b4
e	
b	
b2


ab33	
ab3
b3
ab4
a
ab
ab2	
b3
b4	
e
b


ab4
ab4
b
a
ab
ab2
ab3
b2
b3
b4
	e




Home

Next