Chapter 16

Give an example of two non-isomorphic groups with isomorphic chief series.

Find all the characteristic subgroups of the symmetric group \(S(3)\), the dihedral group \(D(4)\) and the quaternion group of order 8 .

Give an example of an integer \(n\) for which there are two groups of order \(n\) with composition series of different lengths.

Let \(H\) be a characteristic subgroup of \(G\) and \(K\) be a characteristic subgroup of \(H\). Show that \(K\) is a characteristic subgroup of \(G\)