Let
G be a finite group and H a subgroup of G. Let a be an element of G
and aH = {ah : h is an element of H} be a left coset of H. If b is
an element of G as well and the intersection of aH bH is non-empty
then aH and bH contain the same number of elements in G. Thus
conclude that the number of elements in H, o(H), divides the number
of elements...
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
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).
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.
(a) Let G be a finite abelian group and p prime with p | | G |.
Show that there is only one p - Sylow subgroup of G. b) Find all p
- Sylow subgroups of (Z2500, +)
Let D3 be the symmetry group of an equilateral triangle. Show
that the subgroup H ⊂ D3 consisting of those symmetries which are
rotations is a normal subgroup.