# Sylow P-Subgroups, Isomorphisms and Solvable Groups

1. 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.

2. 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.

3. 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

4. Let G be finite. Show that G has a unique largest solvable normal subgroup.

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#### Solution Summary

Sylow P-Subgroups, Isomorphisms and Solvable Groups are investigated.