Which of the following cannot be a class equation of a group of order $10$?
- $1+1+1+2+5=10$
- $1+2+3+4 =10$
- $1+2+2+5 =10$
- $1+1+2+2+2+2=10$
As I can see options 2, 1 and 4 are not class equations as $3$ does not divide $10$ and for 4, $|Z(G)|=2$ will make the group abelian and the equation absurd. So the only possibility is option 3. But Iam not sure whether there could be other arguments besides the ones I have used to dismiss 1, 2, 4 that could dismiss also 3.
Best Answer
There's only one nonabelian group of order $10$: \begin{align*} D_5 = \langle a, x:\, a^5 = 1, x^2 = 1, xa = a^{-1}x\rangle. \end{align*} A brief computation shows that the conjugacy classes of $D_5$ are \begin{gather*} \{1\} \\ \{a, a^4\} \\ \{a^2, a^3\} \\ \{x, ax, a^2x, a^3x, a^4x\}, \end{gather*} which matches item (3) in the problem.