Using Kurosch's subgroup theorem for free proucts,prove that
every finite subgroup of the free product of finite groups is
isomorphic to a subgroup of some free factor.
Let N be a normal subgroup of the group G.
(a) Show that every inner automorphism of G defines an
automorphism of N.
(b) Give an example of a group G with a normal subgroup N and an
automorphism of N that is not defined by an inner automorphism of
G
Suppose G = Z_2 x Z_4. Find the normal subgroups where N is a
normal subgroup of G and H is a normal subgroup of G
s.t. N is isomorphic to H but G/N is not isomorphic to G/H.
1. Let N be a normal subgroup of G and let H be any subgroup
of G. Let HN = {hn|h ∈ H,n ∈
N}. Show that HN is a subgroup of G, and is the smallest
subgroup containing both N and H.