APP. B] ALGEBRAIC SYSTEMS 463
B.53. Consider the symmetric groupS 4. Letα=(
1234
3421)
andβ=(
1234
2431)
.(a) Findαβ,βα,α^2 ,α−^1. (b) Find the orders ofα,β, andαβ.
B.54. Prove the following results for a groupG.
(a) The identity elementeis unique.
(b) EachainGhas a unique inversea−^1.
(c) (a−^1 )−^1 =a,(ab)−^1 =b−^1 a−^1 , and, more generally,(ara 2 ...an)=a−n^1 ...a− 21 a− 11.
(d) ab=acimpliesb=c, andba=caimpliesb=c.
(e) For any integersrands, we havearas=ar+s,(ar)s=ars.
(f) Gis abelian if and only if(ab)^2 =a^2 b^2 for alla, b∈G.
B.55. LetHbe a subgroup ofG. Prove: (a)H=Haif and only ifa∈H. (b)Ha=Hbif and only ifab−^1 ∈H,
(c)HH=H.
B.56. Prove Proposition B.5:AsubsetHof a groupGis a subgroup ofGif: (i)e∈H, (ii) for alla,b∈H, we have
ab,a−^1 ∈H.
B.57. LetGbe a group. Prove:
(a) The intersection of any number of subgroups ofGis a subgroup ofG.
(b) For anyA⊆G,gp(A)is equal to the intersection of all subgroups ofGcontainingA.
(c) The intersection of any number of normal subgroups ofGis a normal subgroup ofG.
B.58. SupposeGis an abelian group. Show that any factor groupG/His also abelian.
B.59. Suppose|G|=p, wherepis a prime. Prove: (a)Ghas no subgroups exceptGand {e}. (b)Gis cyclic and every
elementa=egeneratesG.
B.60. Show thatG={1,−1,i,−i} is a group under multiplication, and show thatG∼=Z 4 by giving an explicit isomorphism
f:G→Z 4.
B.61. LetHbe a subgroup ofGwith only two right cosets. Show thatHis normal.
B.62. LetS=R^2 , the Cartesian plane. Find the stabilizerHaofa=( 1 , 0 )inSwhereGis the following group acting onS:
(a) G=Z×ZandGacts onSbyg(x,y)=(x+m, y+n)whereg=(m, n). That is, each elementginGis a
translation ofS.
(b) G=(R,+) andGacts onSbyg(x, y)=(xcosg−ysing,xsing+ycosg). That is, each element inGrotatesS
about the origin by an angleg.
B.63. LetSbe the regular polygon withnsides, and letGbe the group of symmetries ofS.
(a) Find the order ofG.
(b) Show thatGis generated by two elementsaandbsuchthatan=e,b^2 =e, andb−^1 ab=a−^1 .(Gis called the
dihedral group.)
B.64. Suppose a groupGacts on a setS, say by the homomorphismψ:→PERM(S).
(a) Prove that, for anys∈S: (i)e(s)=s, and (ii) (gg′)(s)=g(g′(s))whereg, g′∈G.
(b) The orbitGsof anys∈Sis defined byGs={g(s)|g∈G}. Show that the orbits form a partition ofS.
(c) Show that∣∣
Gs∣∣
=the number of cosets of the stabilizerHsofsinG. (RecallHs={g∈G|g(s)∈s}.)
B.65. LetGbe an abelian group and letnbe a fixed positive integer. Show that the functionf:G→Gdefined byf(a)=an
is a homomorphism.
B.66. LetGbe the multiplicative group of complex numberszsuch that|z|=1, and letRbe the additive group of real
numbers. ProveG∼=R/Z.
B.67. SupposeHandNare subgroups ofGwithNnormal. Show that: (a)HNis a subgroup ofG. (b)H∩Nis a normal
subgroup ofH. (c)H /(H∩N)∼=HN/N.
B.68. LetHandKbe groups. LetGbe the product setH×Kwith the operation
(h, k)∗(h′,k′)=(hh′,kk′).
(a) Show thatGis a group (called thedirect productofHandK).
(b) LetH′=H×{e}. Show that: (i)H′∼=H; (ii)H′is a normal subgroup ofG; (iii)G/H′∼=K.