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 defined 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 definition 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
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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 afinite 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 definition,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)Reflexive: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.
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