Theorem 2.1. Cauchy’s Theorem: Abelian Case: Let G be a finite
abelian group and p be...
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.
(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, +)
Throughout this question, let G be a finite group, let p be a
prime, and suppose that H ≤ G is such that [G : H] = p.
Let G act on the set of left cosets of H in G by left
multiplication (i.e., g · aH = (ga)H). Let K be the set of elements
of G that fix every coset under this action; that is,
K = {g ∈ G : (∀a ∈ G) g · aH...
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 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 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.)
Let a be an element of a finite group G. The order of a is the
least power k such that ak = e.
Find the orders of following elements in S5
a. (1 2 3 )
b. (1 3 2 4)
c. (2 3) (1 4)
d. (1 2) (3 5 4)