The only partitions of S which are stable are the two partitions mentioned above

GIN G'IN',

PRIM 1. The only partitions of S which are stable are the two partitions mentioned above

PRIM 2. IfH isthe

isotropy

groupofanelement ofS,^{thenHisamaximal}

subgroup

ofG.

Thesetwoconditionsdefinewhat is knownas a

primitive

group,^ormore

accurately,

^a

primitive operation

^{of GonS.}

Instead of

saying

^{that the}

operation

^ofagroup G is2-transitive,^onealso says that it is

doubly

transitive,

47. Let a finite group G operate

transitively

^and

faithfully

^{on a set S with at least 2}

elements and let H be the

isotropy

group of^some element s of S.

(All

^{the other}

isotropy

groups ^are

conjugates

^{ofH.)Prove the}

following:

(a) ^{G is}

doubly

transitive if and

only

^{if Hacts}

transitively

^onthe

complement

ofs in S.

(b) ^{G is}

doubly

transitive if and

only

^{if G =HTH, where Tisa}

subgroup

^{of G} oforder 2 notcontained in H.

(c) ^{If G is}

doubly

transitive, and(G ^:H) ⁼ n, then

#(G) ⁼ den ^- l)n,

where d is the orderof the

subgroup fixing

^{two elements.} Furthermore, H isamaximal

subgroup

^ofG, ^{Le. Gis}

primitive.

48. Let G be a group

acting transitively

^{on asetS with at} least 2 elements. For each

x E G let

I(x)

⁼ number of elements of S fixed

by

^{x. Prove:}

(a)

L I(x)

⁼ #(G).

XEG

(b) ^Gis

doubly

^{transitiveif and}

only

^if

L f(X)2

^{= 2} #(G).

XEG

49. A group^asan

automorphism

group. Let G be^agroup and let

Set(

G)^bethecategory ofG-sets(Le. ^setswitha

G-operation),

^{Let F:}

Set(G)

^{Setbe the}

forgetful

^functor,

whichto each G-set

assigns

^{the set itself. Show that}Aut(F)^is

naturally isomorphic

toG.

Fiber

products

^{and co}

products

Pull-backsand

push-outs

50. (a) Show that fiber

products

exist in thecategory of abelian groups. In fact, ^IfX, ^Y

are abelian groups with

homomorphisms f:

^X-+Z and g: Y^-+Z show that X ^xzYis theset ofall

pairs

(x,

y)

^{with x EX} and y^EYsuch that

f(x)

⁼

g(y).

The maps Pt, P2^arethe

projections

^on the first and second factor

respectIvely.

(b)

Show that the

pull-back

^ofa

surjectIve homomorphism

^is

surjective.

51.

(a)

Show that fiber

products

^{exist inthe}category^ofsets.

(b)

In any category e, consider the category ^e7. of

objects

^{overZ. Let h: T-+Z}

beafixed

object

^{in this}category, ^{Let Fbethe}functorsuch that F(X)⁼

Morz(T, X),

where X isan

object

^overZ,^{and Mor}zdenotes

morphisms

^{overZ. Show that}

F transforms fiber

products

^{over Z into fiber}

products

^{in the}category of^sets.

(Actually,

^onceyou have understood thedefinitions,^{this is}

tautological.)

52,

(a)

^{Show that}

push-outs (i.e.

^fiber

coproducts)

exist in thecategory of abeliangroups.

Inthiscasethe fiber

coproduct

^oftwo

homomorphisms f,

g^asabove isdenoted

by

^X(f)z Y. Show that it is the factor group

Xz ^{y =} (X

Y)/W,

where Wis the

subgroup consisting

of all elements

(f(z),

^-

g(z»

^withzEZ.

(b)

^Showthatthe

push-out

^ofan

injective homomorphism

^is

injective.

Remark. Afteryou have read about modules over

rings,

you should ^notethat the abovetwoexercises

apply

^{tomodulesaswellastoabelian}groups,

53. Let H, G,^G'be groups, and let

f:

^H-+G, g^:H^-+ G'

be two

homomorphisms.

^{Define the notion of}

coproduct

^{ofthese two homomor-}

phisms

^overH,andshow that it exists.

54. (Tits). ^{Let G be a} group and let

{GJiEl

^{be a}

family

^of

subgroups generating

^G.

Suppose

^G operates^{on aset S. Foreach i E} I, suppose

given

^asubset

Si

^ofS, ^and let sbea

point

^{of S -}

l) Si.

Assume that for each 9^E

G;

^-

{e},

^wehave

,

gSj

^C

^S;

^for

allj

^{=1= i, and} g(s) ^E

S;

^{for all i.}

Prove that G is the

coproduct

^{of the}

family {GJ;El'

(Hint:

Suppose

^a

product

g. ^... gm ⁼ idonS,

Apply

^this

product

^{to s, anduse}

Proposition

12.4.) 55. Let M E GL₂(C) (2 ^{x 2}

complex

matrices with non-zerodeterminant). ^{We let}

(

^{a b}

)

^{az + b}

M= ,andforzECweletM(z)=

d

'

c d ^{cz +}

Ifz ⁼

-d/

^c (c ⁼¹⁼ 0)^{then we} putM(z) ^{= 00, Then}you ^can

verify

(and you should have seen

something

like this in a course in

complex analysis)

^{that GL}2(C) ^thus operates^{on C U}

{oo}.

^{Let A, A' be the}

eigenvalues

^{of Mviewed as a}linear map^on C2. Let W, ^{W' be the}

corresponding eigenvectors,

W⁼

f(W., w2)

^{and W' =}

f(W;, w;),

By

^afixed

point

^{of MonCwe mean a}

complex

^{numberzsuchthat}M(z)^{= z.}Assume that M hastwodistinct

fixed

points ^{=1= 00.}

(a) Show that there cannot be more than ^two fixed

points

and that these fixed

points

^{are w =}

wllw2

^{and w' =}

wi/w2.

^{Infact one}may take W⁼ t(w, 1), ^{W' =} t(w', 1).

(b) ^{Assume that}

1 AI

^<

1

^A'

I,

^{Givenz =1= w, show that}

lim

Mk(z)

^{= w'.}

k-oo

[Hint: Let S ⁼ (W, W') ^andconsider

S-IMkS(Z)

^{= exkzwhere ex =}

AI A'.]

56. (Tits) ^Let

M.,

^{..., M}r E GL₂

(C)

^{be a}finite number of matrices. Let

A;, A;

^{be the}

eigenvalues

^of

M;.

Assume that each

M;

^{has twodistinct}

complex

^fixed

points,

^and

that

I A; I

^<

1 A; I.

^{Also assume that the fixed}

points

^{forMI'.,., Mr are all distinct}

from each other, Prove that there exists a

positive integer

k such that

M,

^.,,,

M

are the freegenerators^ofafree

subgroup

^{of GL}2(C), [Hint:Let wi'

w;

^{bethe fixed}

points

^of

M;.

^Let

V;

^{be asmall} disc centeredat Wi and

V;

^{a small} disc centered at

w;.

^Let

S;

⁼

V;

^U

V;.

^{Let sbe a}

complex

^{numberwhich does not}lie in any

S;.

^Let

G;

⁼

(M).

Show that the conditions of Exercise 54 are satisfied for k

sufficiently large.]

^.

s.

57. Let Gbeagroup

acting

^{on asetX. Let Y beasubsetofX. Let G}ybe thesubsetof G

consisting

^{of those elements}g such that_{g Y n} Yis not empty. ^{Let G}y be the

subgroup

^{of G}

generated by

^{Gy. Then G}

yY

^and (G ^{- G}

y)Y

^are

disjoint. [Hint:

Suppose

that there existgl EG_yand_{g2 E} Gbut_g2

$

Gy,and elementsYI, Y2, E Y such that_g2Yl ⁼_{g2Y2. Then}

g:;lglYI

⁼ Y2, ^so

g:;lg)

^{E G}ywhence_{g2 E} Gy,contrary

to

assumption.]

Application. Suppose

^thatX= GY,^{butthatXcannotbe}

expressed

^{as a}

disjoint

unionasabove unlessoneof thetwo setsisempty.^Thenweconclude that G ^- G_y isempty,^andtherefore Gygenerates G.

Example

^1.

Suppose

^{Xisaconnected}

topological

space, ^Yis open, and G^acts

continuously.

Then all translates of_{Yare open,}soGis

generated by

^Gy.

Example

^2.

Suppose

^{G isa} discrete group

acting continuously

^and

discretely

onX.

Again

suppose^Xconnected and Yclosed._{Then any} union of translates ofY

by

elements of G isclosed,^so

again

^{G - Gyis}empty, ^{and G}ygenerates G.

CHAPTER II