Which of the following cannot be class equation of following order of group?
A. $12=1+3+4+4$
B. $24=1+6+8+6+3$
C. $39=1+3+6+3+13+13$
D. $4=1+1+2$
I have checked that (B) cannot be the right option since it is precisely the class equation of $S_4$.
My question is, given an integer and any partition of it, is there a general method or tricks to figure out whether it can be the class equation of any group of order equal to that integer (without actually classifying all groups of that order and checking their equations)? In other words, if you are given a more general question of this sort, how would you go about solving it or at least eliminating some options?
Best Answer
A. This Is the class equation of the alternating group $A_4$.
C. Order of a conjugacy class must divide order of the group (from Orbit-Stabilizer Theorem). But here $6$ does not divide $39$. So, this is not a valid class equation.
D. There are only two groups of order 4 and they are abelian. So $|Z(G)| = 4$.