Which of the following equations can occur as the class equation of a group of order 10?

Concept:

If G be a group such that o(G) = 2p where p is odd prime then G ≅ \(\mathbb Z_{2p}\) or G ≅ \(\mathbb D_p\)

(ii) If o(G) = n where G is abelian group then class equation of G is n = 1 + 1 + ... + 1 (n times)

Explanation:

Given o(G) = 10 = 2.5

So here p = 5 which is odd prime.

Hence G ≅ \(\mathbb Z_{10}\) or G ≅ \(\mathbb D_5\)

If G ≅ \(\mathbb Z_{10}\) then it is cyclic so abelian. Therefore class equation of a group of order 10 is 10 = 1 + 1 + … + 1 (10-times)

If G ≅ \(\mathbb D_5\) then there will be 5 rotation and 5 reflection.

So in this case class equation of a group of order 10 is 

10 = 1 + 2 + 2 + 5

Hence option (1) is correct