Assignment:
Q1. a) If M and N are normal subgroups of G then G/M is isomorphic to a subgroup of G/M x G/N.
b) If G/M and G/N are solvable, then G/(M intersect N) is solvable.
Q2. Let G be finite and P be a Sylow p subgroup of G. Suppose the normalizer of P in G is a subset of H is a subset of G. Show that the normalizer of H in G is equal to H.
Q3. Let G be finite and solvable and suppose M(less than)G is a maximal subgroup and the core of M in G is equal to 1. Let N be a minimal normal subgroup of G. Show the following:
a) NM=G and (N intersect M)=1
b) the centralizer of N in M is equal to 1
c) N=the centralizer of N in G
d) N is the unique minimal normal subgroup of G
Q4. Let G be finite. Show that G has a unique largest solvable normal subgroup.
Provide complete and step by step solution for the question and show calculations and use formulas.