Let G be a finite group, and let H be a subgroup of G then Which of the following statement is/are true?
1
If G is a cyclic group of order n , and H is a subgroup of G , then H is normal in G if and only if H is generated by \(g^k \) , where g is a generator of G and k divides n .
2
If G is cyclic of order 2n , then the quotient group \(G/\langle g^2 \rangle \) is isomorphic to \(\mathbb{Z}_2 \) .
3
If G is cyclic and N is a normal subgroup of G , then N is necessarily cyclic and its order divides the order of G .
4
Every normal subgroup of a cyclic group is cyclic, but the converse does not hold for non-abelian groups.