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40 votes

The following is the incomplete operation table of a $4-$element group.

$$\begin{array}{|l|l|l|l|l|} \hline \textbf{*} & \textbf{e}& \textbf{a} &\textbf{b} & \textbf{c}\\\hline \textbf{e} & \text{e}& \text{a} & \text{b} & \text{c} \\\hline \textbf{a} & \text{a}& \text{b} & \text{c} & \text{e}\\\hline \textbf{b}\\\hline \textbf{c} \\\hline\end{array}$$

The last row of the table is

  1. $c\;a\;e\; b$
  2. $c\; b\; a\; e$
  3. $c\; b\; e\; a$
  4. $c\; e\; a\; b$
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4 Answers

Best answer
54 votes
54 votes
  1. Group of order Prime Square $(p^2)$ is always abelian. See here 
  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.

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41 votes
41 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.
4 votes
4 votes

__ __ __ __

 

Inverse of a is c. So, inverse of c is a.

So, we have

__ e __ __

Only Option D satisfies.


Alternatively

 

In some Cayley tables, you'd notice this pattern that the elements are being rotated by 1 position to the left.

First row: $e,a,b,c$

Second row: $a,b,c,e$

Third row: $b,c,e,a$

Fourth row: $c,e,a,b$

0 votes
0 votes

It is given that the given set of 4 elements is group.

The element ’e’ is clearly identity as the row corresponding to it has all values same as the other operand.

 

Also, since a*c is e, c*a should also be e which is only the case in option D

Answer:

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