Which of the following cannot be the class equation of a group G of order 10?
1)1+1+1+2+5.
2)1+2+3+4.
3)1+2+2+5.
4)1+1+2+2+2+2.
Reasoning
(1) is not the required class equation because if it is then the order of centre of G will be not possible due to Lagrange's theorem.
(4) is not the class equation because if it is so then the index of centre of G
in G will be 2 which implies G to be abelian leading to the class equation 10.Which is not possible.
In answer key correct options are (1),(2),(4).I'm not getting why (2) is not the class equation.
In my class notes it is told that $S_3$ is the largest group with each conjugacy class of distict sizes.But,G is a group of order 10 with each conjugacy class of distict size in (2).
Is there any other reason why (2) is not the class equation?
Best Answer
1+2+3+4 means there are four distinct conjugate classes having order 1,2,3 & 4. Since, |cl(a)|.|N(a)|=|G|, then for class of order 3, 3.|N(a)|=10. But 10 is not multiple of 3.Hence the class equation is not possible.