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+16 votes
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The following is the incomplete operation table of a 4-element group.

* e a b c
e e a b c
a a b c e
b        
c        

The last row of the table is

  1. c a e b
  2. c b a e
  3. c b e a
  4. c e a b
asked in Set Theory & Algebra by Veteran (69k points) | 777 views
Although Rajesh's answer is best way to approach this problem , yet these is another approach.

We can observe given group is cyclic group , where 'a' is generator , and every cyclic group is abelian too. So fill the missing entries on the bases of commutative property of abelian groups

$a^{1}=e$

$a^{2}=a*a=b$

$a^{3}=a*(a*a)=a*b=c$

$a^{4}=a*a*(a*a)=a*(a*b)=a*c=e$

This might help ...

2 Answers

+11 votes
Best answer
1.Group of order Prime Squared (p^2)is always Abelian group.See

2. If the group is abelian, then x * y = y * x (Commutative)for every x,y in our group. Therefore, the i-j th entry is equal to the j-i th entry in the Cayley table, so the table is symmetric.(Informally 1st row is same as 1st col. 2nd row is same as 2nd col..and so on)

 

->>Here order 4=p2=22  (p=2),Hence it is abelian group.

->>now abelian groups cayle table is symmetry so

1st row will 1st col and 2nd row will 2nd col..

which match with option D only.

answered by Veteran (23.4k points)
selected by
O(g) <= 5  will always be abelian therfore symmetric property can be applied in cayley table
+22 votes
From First row you can conclude that e is the identity element.

=> Using the above fact, from second row you can conclude that a and c are inverses of each other.

=> In fourth row:

First element : c*e = c (e is identity)

Second element : c*a = e (inverse)

Option 4 matches this.
answered by Loyal (4.3k points)

Row1 tells e is left identity, to be an identity element it should be right as well as left identity.

in first column we see a*e = e, this means e is right identity. therefore e is identity.  and identity element is unique hence e is the only identity.

This gives - 

*

e

a

b

c

e

e

a

b

c

a

a

b

c

e

b

 b

 

 

 

c

 c

 

 

 

Property of Cayley Table:-  it can not contain any element twice in any row or column. (see here)

 

using these two properties we can fill every entry above.

 

How did you conclude that it is a Cayley table?
Hello mojo-jojo

I'm removing Best Tag from your answer. An answer should be an explanation. exam oriented solution doesn't matter.


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