Homework2

Solutions to Homework Two

1.Let f be a homomorphism from group G to group G′,for any a∈G,whether or not a and f(a)have the same order?

Solution.It doesn’t always hold.For example,let f be a homomorphism from G=(Z,+)to G′={0},for any0=m∈Z,?(m)=∞,but?(f(m))=1.

2.(a)Let H be a subgroup of G and let g∈G.The conjugate subgroup gHg?1is de?ned to be the set of all conjugates ghg?1, where h∈H.Prove that gHg?1is a subgroup of G.

(b)Prove that a subgroup H of a group G is normal if and only if gHg?1=H for all g∈G.

Proof.(a)1=g1g?1∈gHg?1.For any h,k∈H,(ghg?1)?1= gh?1g?1∈gHg?1and(ghg?1)(gkg?1)=g(hk)g?1∈gHg?1.Thus gHg?1is a subgroup of G.

(b)Assume that H is normal,then for any g∈G,h∈H, ghg?1∈H.So gHg?1?H.Conversely,for any h∈H,g?1hg∈H, which implies h=g(g?1hg)g?1∈gHg?1.Thus gHg?1=H for all g∈G.

Assume that gHg?1=H for all g∈G.From the de?nition of the normal subgroup,we have H is normal.

3.Prove that if a group contains exactly one element of order2, then that element is in the center of the group.

Proof.Let a is the exact element of order2in G.For any element b∈G,we consider the element bab?1.Obviously,we have

bab?1=e(otherwise a=e),but(bab?1)2=ba2b?1=e.Thus bab?1=a,i.e.,ab=ba.So a is in the center of G.

4.If A,B are subsets of a?nite group G such that|A|+|B|>|G|, prove that G=AB.

Proof.We only need to prove that G?AB since AB?G is clear.Set B?1={b?1|b∈B}.For any g∈G,since|gB?1|=|B|, we have|A|+|gB?1|>|G|.This implies that A∩gB?1=?.Let a∈A∩gB?1,thus a=gb?1for b∈B,that is,g=ab∈AB.

5.(a)Prove that the relation a conjugate to b in a group G is an equivalence relation on G.

(b)Describe the elements a whose conjugacy class(=equivalence class)consists of the element a alone.

Proof.(a)(i)Transitive:Suppose that a conjugate to b and b conjugate to c.By de?nition,there exists g,h∈G such that gag?1=b and hbh?1=c.Then we have(hg)a(hg)?1=hbh?1=c, which means a conjugate to c.

(ii)Symmetric:Suppose that a conjugate to b,then there exists g∈G such that gag?1=b,thus a=g?1bg,that is,b is conjugate to a.

(iii)Re?exive:For any a∈G,one always has1a1=a.

From the above discussion,we know that the relation is an equiv-alence relation on G.

(b)The conjugacy class of a is{a}if and only if a∈Z(G),the center of group G.