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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 (59.5k points) | 933 views
+3
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

$a^{1}=e$

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

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

$a^{4}=a*a*(a*a)=a*(a*b)=a*c=e$
+1

This might help ...

2 Answers

+14 votes
Best answer
  1. Group of order Prime Square $(p^2)$ is always abelian. See
  2. If the group is abelian, then $x * y = y * x$ for every $x,y$ in it (Commutative). Therefore, the $(i,j)$ entry is equal to the$( j,i)$ entry in the Cayley table making the table is symmetric. $($Informally, $1^{st}$ row is same as $1^{st}$ column, $2^{nd}$ row is same as $2^{nd}$ column and so on$)$

Here, order $4=p^2=2^2  (p=2).$ Hence, it is abelian group.

Now abelian group's Cayley  table is symmetric. So, $1^{st}$ row will be same as $1^{st}$ col and $2^{nd}$ row will be same as $2^{nd}$ column.

Matches with option D only.

answered by Boss (22.7k points)
edited by
+1
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 Active (4k points)
+10

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.

0
How did you conclude that it is a Cayley table?
0
Hello mojo-jojo

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

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