Let D3 be the symmetry group of an equilateral triangle. Show
that the subgroup H ⊂...
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.
4.- Show the solution:
a.- Let G be a group, H a subgroup of G and a∈G. Prove that the
coset aH has the same number of elements as H.
b.- Prove that if G is a finite group and a∈G, then |a| divides
|G|. Moreover, if |G| is prime then G is cyclic.
c.- Prove that every group is isomorphic to a group of
permutations.
SUBJECT: Abstract Algebra
(18,19,20)
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 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).
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.
Find two distinct subgroups of order 2 of the group D3 of
symmetries of an equilateral triangle. Explain why this fact alone
shows that D3 is not a cynic group.
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
Three rods each of mass m and length I are joined together to form an equilateral triangle as shown in figure. Find the moment of inertia of the system about an axis passing through its centre of mass and perpendicular to the plane of triangle.