北京邮电大学 离散数学下 群论 作业错题讲解(12)

北京邮电大学离散数学群论作业详解

群论9.4 (2)28,32@page 349; 24@page338.

24-Let A={0,1} and consider (A*, ?), (N,+).

(1)f(α)=length(α), show f:A*->N is homomorphism.

f(α?β)=f(αβ)=length(α)+length(β)=f(α)+f(β)

(2)R: f(α)=f(β),show R is congruence relation.

First show R is equivalence relation.

2. if f(α)=f(β) and f(χ)=f(δ),then f(α?χ) = f(β?δ)

(3)A*/R and N is isomorphic.

(A*/R,*) : [α]*[β]=[α?β].

Let g([α])=length(α), (1)show g: A*/R->N is homomorphism. g([α] *[β])=g([α?β])=length(α)+length(β)=g([α])+g([β])

(2) onto : ?x∈N, let α=00…0(x factors) ,g([α])=x.

(3) one-to-one : suppose g([α])=g([β]),

length(α)=length(β),so f(α)=f(β), then αRβ, [α]=[β].