抽象代数作业答案群group

1. Compute the order of β=(13)(247) and γ=α-1βα, where α=(36)(254).

2. Special Orthogonal matrices SO(2,R) is a set of elements Aθ in the form below, equipped with matrix production.

1)Show SO(2,R) is an Abelian group;

2)Give an appropriate definition of Aθk by matrix production, and prove

Aθk= A kθ, for k∈Z;

3) Find all the elements of finite order in SO(2,R).

3. GL(2,Q) denotes the set of the nonsingular (having nonzero determinant, 行列式非0) rational square matrix of length 2 under matrix product.

1) show that it’s a group;

2) compute the order of A=[0, -1; 1, 0] and B=[0, 1; -1, 1] ;

3) what’s the order of AB ?

4) GL(k,R) denotes the set of the nonsingular k×k real matrix under matrix product. Show that it is a group.

4. 1) The affine set Aff(1,R) consists of all functions R→R of the form

f a,b(x)=ax+b, where a, b are fixed real numbers with a≠0. Show that Aff(1,R) is a group under the function composition;

2) The affine set Aff(k,R) consists of all functions R k→R k of the form

f A,b(X)=AX+b, where A is a k×k matrix with |A|≠0, b is a k×1 vector. Show that Aff(k,R) is a group under the function composition.

1. If finite groups H and K are subgroups of a group G and if

gcd(|H|,|K|)=1, prove that H∩K={1}.

2. Let G be a group of order 4. Prove that either G is cyclic or x2 = 1 for every x∈G, and G must be abelian.

3. Let G be a finite abelian group and |G| is odd, prove that for every x∈G, there exists exactly one g∈G with g2=x.

4. If f: G→H is a homomorphism and x∈G has order k, prove that f(x)∈H has order m, where m|k.

5. If f: G→H is a homomorphism and if gcd(|G|,|H|)=1, prove that f(x)=1 for all x∈G.

6. Let H and K be groups. H*={(h,1): h∈H}, K*= {(1,k): k∈K}.

1) Prove that H* is a normal subgroup of H×K.

2) Prove that (H×K)/K*≌H.