MATH数学作业代写-MT4516代写-Finite Mathematics代写

School of Mathematics and Statistics

MT4516 Finite Mathematics

MATH数学作业代写 By analogy with (i) defifine the direct product of two vector spaces over a fifield F and prove that it is again a vector space over F.

1.(i) For two groups G and H let G × H = {(g, h) : g ∈ G, h ∈ H}, and defifine the multiplication in this set by (g, h)(g1, h1) = (gg1, hh1). MATH数学作业代写

Prove that G × H is a group. (G × H is called the direct product of G and H.)

(ii) By analogy with (i) defifine the direct product of two vector spaces over a fifield F and prove that it is again a vector space over F.

(iii) Defifine the direct product of two fifields using componentwise oper-ations. Is it a fifield?

2.Let G be an abelian group with identity 1, and let H be a subgroup of G.

(Thus H ⊆ G and H is a group with respect to the same multiplication.) A coset of H is a subset of G of the form aH = {ah : h ∈ H} with a ∈ G.(i) Prove that H = 1H is a coset of itself and that a ∈ aH for every a ∈ G.

(ii) Prove that the mapping f : H −→ aH defifined by f(x) = ax is a bijection and conclude that any two cosets of H have the same number of elements.

(iii) Prove that any two cosets of H are either disjoint or else are equal. MATH数学作业代写

(iv) Prove that G is partitioned by the cosets of H, and that if G is fifinite then the order of H divides the order of G.

3.Let F be the set of all constant and linear polynomials over the fifield Zp. Defifine the addition in F to be the ordinary addition of polynomi-als (modulo p), and defifine the multiplication in F to be the ordinary multiplication of polynomials (modulo p) with the additional rule that x 2 = αx + β, so that x 2 can be replaced by αx + β whenever it occurs. Prove that if p = 3 and α = β = 1 then F is a fifield, while if p = 2, α = 0 and β = 1 then F is not a fifield.