I got stuck with two questions dealing with characteristic groups
the first - G is a finite group characteristically simple and N is a minimal normal sub group in G (contains no proper normal subgroups of G). if is a direct sum of isomorphic copies of where k is maximal, then it is normal in G.
I can show that N is characteristically simple, and also all its isomorphic copies. I tried to show that if then , and show that either there is another copy of N that I can join to the sum, or H is normal.
The problem is I can't see how I can use the fact that G is characteristically simple. I know that Z(G) and the commutator subgroups are characteristic in G but couldn't find what to do with it.
The second question is to show that if N is normal in G then where is the intersection of all maximal groups in A. I thougt to use and show that there is an intersection of some maximal groups in N that gives it. another thing is if H is maximal in G, and is not contained in H then maybe will give a contradictory to the maximality of H (which is well defined bacuase char N normal in G).