Question

Let G be a group. Suppose that the class equation for G is 1 + 4 + 5 + 5 + 5. (a) Does G have a subgroup of order 5? If so, then is it normal in G? Prove your answers. (b) Does G have a subgroup of order 4? If so, then is it normal in G? Prove your answers.

Answer 1

Answer: (a) G have a subgroup of order 5 but it is not normal in G.

(2) G have a subgroup of order 4 but it is normal in G.

Explanation:

Since we have given that

the Class equation of G is 1+4+5+5+5.

So, the order of G i.e. |G|=1+4+5+5+5=20

Since we can see that there are three conjugacy classes of order 5.

(a) So, G have a subgroup of order 5.

Since number of subgroups of order 5 =3

But 3 does not divide 20.

so, it is not normal in G.

(b) G have a subgroup of order 4.

Since there are only one subgroup of order 4.

and 1 divides 20

so, it is normal in G.