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.
Let G be a group and let N ≤ G be a normal subgroup.
(i) Define the factor group G/N and show that G/N is a
group.
(ii) Let G = S4, N = K4 = h(1, 2)(3, 4),(1, 3)(2, 4)i ≤ S4. Show
that N is a normal subgroup of G and write out the set of cosets
G/N.
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
(a) Suppose K is a subgroup of H, and H is a subgroup of
G.
If |K|= 20 and |G| = 600, what are the possible values for
|H|?
(b) Determine the number of elements of order 15 in Z30 Z24.
G
is a group and H is a normal subgroup of G. List the elements of
G/H and then write the table of G/H.
1. G=Z10, H= {0,5}. (Explain why G/H is congruent to Z5)
2. G=S4 and H= {e, (12)(34), (13)(24), (14)(23)
Let G be a group and K ⊂ G be a normal subgroup. Let H ⊂ G be a
subgroup of G such that K ⊂ H Suppose that H is also a normal
subgroup of G. (a) Show that H/K ⊂ G/K is a normal subgroup. (b)
Show that G/H is isomorphic to (G/K)/(H/K).