Let G be an abelian group and K is a subset of G.
if K is a subgroup of G , show that G is finitely generated if
and only if both K and G/K are finitely generated.
Let (G,+) be an abelian group and U a subgroup of G. Prove that
G is the direct product of U and V (where V a subgroup of G) if
only if there is a homomorphism f : G → U with f|U =
IdU
Let G be an abelian group.
(a) If H = {x ∈ G| |x| is odd}, prove that H is a subgroup of G.
(b) If K = {x ∈ G| |x| = 1 or is even}, must K be a subgroup of G?
(Give a proof or counterexample.)
(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, +)
Theorem 2.1. Cauchy’s Theorem: Abelian Case: Let G be a finite
abelian group and p be a prime such that p divides the order of G
then G has an element of order p.
Problem 2.1. Prove this theorem.
Let G be an abelian group and n a fixed positive integer. Prove
that the following sets are subgroups of G.
(a) P(G, n) = {gn | g ∈ G}.
(b) T(G, n) = {g ∈ G | gn = 1}.
(c) Compute P(G, 2) and T(G, 2) if G = C8 ×
C2.
(d) Prove that T(G, 2) is not a subgroup of G = Dn
for n ≥ 3 (i.e the statement above is false when G is...
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)