D4 = {(1),(1, 2, 3, 4),(1, 3)(2, 4),(1, 4, 3, 2),(1,
2)(3, 4),(1, 4)(2, 3),(2, 4),(1, 3)}
M = {(1),(1, 4)(2, 3)}
N = {(1),(1, 4)(2, 3),(1, 3)(2, 4),(1, 2)(3, 4)}
Show that M is a subgroup N; N is a subgroup D4, but
that M is not a subgroup of D4
Show that the groups of the following orders have a normal Sylow
subgroup.
(a) |G| = pq where p and q are
primes.
(b) |G| = paq where p and
q are primes and q < p.
(c) |G| = 4p where p is a prime greater than
four.
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 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)