Show that if G is a nonabelian finite group then ZG

Show that if G is a nonabelian finite group, then |Z(G)| <= |G|/4

Solution

Z(G) the centre of the group is a subgroup of G.

Now Since G is non abelian , G/Z(G) cannot be a cyclic group

otherwise G would be abelian.

So Now we want bound on G/Z(G)

So how smallest can be a non cyclic group?. Non cyclic group must be of order at least 4

hence

|Z(G)|<= |G|/4