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
(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...
Prove that any two groups with one element are isomorphic.
Prove that any two groups with two elements are isomorphic.
Prove that any two groups with three elements are
isomorphic.
Prove that an abelian group G of order 2000 is the direct
product PxQ where P is the Sylow-2 subgroup of G, and Q the Sylow-5
subgroup of G. (So order of P=16 and order or Q=125).
Let the demand and supply functions for a commodity be Q =D(P)
∂D < 0
∂P
Q=S(P,t) ∂S>0, ∂S<0 ∂P ∂t
where t is the tax rate on the commodity.
(a) What are the endogenous and exogenous variables? (b)
Derive the total differential of each equation.
(c) Use Cramer’s rule to compute dQ/dt and dP/dt. (d)
Determine the sign of dQ/dt and dP/dt.
(e) Use the Q − P diagram to explain your results.
Find the Taylor series with n...