Endomorphisms and Automorphisms

In th is c h a p te r we are going to s tu d y th e follow ing s itu a tio n . Let F b e a free g ro u p a n d R a n d S n o rm al su b g ro u p s of F such th a t S is c o n ta in e d in R.

F u rth e rm o re , let J\f be a set of n o rm a l su b g ro u p s of F all of w hich c o n ta in 5 a n d are c o n ta in e d in R. T h e p ro b lem here is th e d e te rm in a tio n of th e different iso m o rp h ism ty p e s of facto r groups F / N for N £ Af . P a r tly in sp ire d by th e id eas of th e p -g ro u p g en e ra tio n alg o rith m ( O ’B rien , 1990) we will d escrib e c o n d itio n s th a t a d m it an actio n on J\f by a c e rta in g ro u p of a u to m o rp h is m s . T h e o rb its u n d e r th is ac tio n co rresp o n d to th e iso m o rp h ism ty p e s of fa c to r g ro u p s F / N for

N e A/\

6.1 Definition Let G be a g roup w ith n o rm al su b g ro u p s M a n d N a n d let e

be an en d o m o rp h ism of G such th a t M e C N. T h e n (M g)e — N ( g e ) defines a h o m o m o rp h ism e from G / M to G / N . T h e h o m o m o rp h ism e is said to b e induced

by e. Let ß be a h o m o m o rp h ism fro m G / M to G / N . T h e e n d o m o rp h ism ß* of

G is said to be a lifting of ß to G if M ß* C N a n d ß* ind u ces ß.

T h ro u g h o u t th is c h a p te r let F be a free g ro u p w ith a free g e n e ra tin g set { x i , . . . , £ n } an d let M an d N be n o rm al su b g ro u p s of F.

Let p be a h o m o m o rp h ism from F / M to F / N a n d { p i , . . . , p n } a set of elem en ts in F such th a t [ Mx ß j p = Ny i . T h e n M w ( x i,. . . , x n ) is m a p p e d by p

m a p s w ( xi ,. . . , xn ) to w ( y j, . . . , y n ). If w ( x\ ,. . . , x n ) is an elem en t of M , th e n

w ( y i, . . . , y n ) is an elem ent of N :

jV = My? = ( M w ( x u . . . ,x„))y? = iV™(y1?. • •

T h is m e an s th a t My?* C N an d y?* is a liftin g of y?.

If 5 is a n o rm a l su b g ro u p of F c o n tain ed in M D N such th a t 5y?* C 5, th e n y?* in d u ces an e n d o m o rp liism y?* on F / S v ia (5x{)y?* = S y t . C learly, y?* m ap s M / 5 in to N / S .

L et y? be an iso m o rp h ism fro m F / M to F / N . It is a n a tu r a l q u estio n to ask if y? h as a liftin g to an a u to m o rp h is m of F / S . T h e follow ing th e o re m can b e used to p ro v id e an answ er to th is q u estio n .

6.2 Theorem (G asclh itz 1955)

If N is a fin ite n o rm a l su b g ro u p of a g ro u p G a n d G can be g e n e ra te d by n elem en ts, th e n for each g e n e ra tin g set { N ,. . . , N g n } of G / N th e re is a g e n e ra tin g set {/ i j , . . . , h n } of G such th a t hi £ Ngi for 1 < i < n.

In o rd e r be able to ap p ly th is th e o re m , it is a ssu m ed th a t N / S is finite a n d th a t y? is an iso m o rp h ism fro m F / M to F / N such th a t every liftin g of y? to F leaves 5 in v a ria n t. T h e la tte r is th e case if 5 is fully in v a ria n t, h e., in v a ria n t u n d e r all en d o m o rp h ism s of F. H ow ever, we do n o t w ant to im p o se th is s tro n g e r co n d itio n on 5. T h e im ages of M x i u n d e r y? for 1 < i < n g en era te

F / N b eca u se y? is an iso m o rp h ism . By G a s c lh itz ’ th e o re m ap p lied to G = F / S ,

elem en ts yi £ ( M i^ y ? can be chosen such th a t such th a t { S y i,. . . , S y n } is a g e n e ra tin g set for F / S . As before, X{ t—» xji defines a liftin g y?* of y? to F w hich in d u c es an en d o m o rp liism y?* of F / S . By choice of th e y{, th is e n d o m o rp h ism is su rjectiv e. It is clear th a t y?* m a p s M / S to N / S . In g en eral, it seem s to be difficult to s ta te co n d itio n s u n d e r w hich y?* is in jectiv e. H ow ever, if th e g ro u p in q u estio n is H opfian th e n th e s u rje c tiv ity of y?* im plies th e in jectiv ity . For th e p re se n t p u rp o se we will assum e th a t th e involved g ro u p s are H o pfian w here necessary. We su m m a rize th e discussion so far in th e follow ing lem m a.

6.3 Lemma Let F , M an d N be as above a n d 5 a n o rm al su b g ro u p of F

-

S

is c o n ta in e d in

M

f l

N,

-

th e q u o tie n t g ro u p

F / S

is hop fian an d -

N / S

is finite.

T h e n for any iso m o rp h ism

p

from

F / M

to

F / N

such th a t

S

is in v a ria n t u n d e r an y liftin g of

p

to

F

th e re is an a u to m o rp h is m of

F / S

th a t in d u ces

p

a n d th a t m a p s

M / S

to

N / S .

If we assu m e th a t

S

is fully in v a ria n t we get th e follow ing corollary.

6.4 Corollary Let 5 be a fully in v a ria n t s u b g ro u p of

F

a n d such th a t

F / S

is H opfian. T h e n tw o fin ite n o rm al su b g ro u p s of

F / S

lie in th e sam e c h a ra c te ristic class if an d only if th e ir co rresp o n d in g q u o tie n t g ro u p s are iso m o rph ic.

P ro o f: If tw o n o rm a l su b g ro u p s of a g ro u p are c o n ju g a te u n d e r an a u to m o rp h is m , th is a u to m o rp h is m ind uces an iso m o rp h ism b etw ee n th e ir q u o tie n t g ro u p s. T h e

converse is tr u e by th e previou s lem m a. ■

Now let

R

be a n o rm al su b g ro u p of

F

c o n ta in in g

M N

a n d su p p o se th a t

S

is in v a ria n t u n d e r all en d o m o rp h ism s of

F

th a t leave

R

in v a ria n t. If (

R / M ) p

=

R / N

, th e n th e liftin g

p*

to an e n d o m o rp h ism of

F

leaves

R

in ­ v a ria n t. U n d er th e co n d itio n s of th e lem m a, th is e n d o m o rp h ism can be chosen such th a t it in d u ces an a u to m o rp h is m

p*

of

F / S .

C learly,

p*

leaves

R / S

in v a ri­ a n t a n d in du ces an a u to m o rp h is m on

F / R

or, vice v ersa,

p*

is a liftin g of an a u to m o rp h is m of

F / R .

6.5 Theorem Let

R

an d 5 be n o rm al su b g ro u p s of F ,

R

c o n ta in in g 5 , such th a t every liftin g of any a u to m o rp h ism of P / i ? to a n e n d o m o rp h ism of

F

leaves

S

in v a ria n t. F u rth e rm o re , assum e th a t

F / S

is H opfian.

L et

Af

b e a set of n o rm al su b g ro u p s of

F

satisfy in g th e follow ing co n d itio n s. - E very

M

£

Af

co n tain s 5 a n d is c o n tain ed in

R.

-

For every

M

E

Af

th e q u o tie n t

M / S

is finite.

- E very iso m o rp h ism fro m

F / M

to

F / N

for M ,

N

£

Af

m a p s

R / M

in to

R / N .

T h e n for

M

a n d

N

in

Af

th e q u o tien t groups

F / M

a n d

F / N

are iso m o rp h ic if a n d only if th e re is an a u to m o rp h is m a of

F / R

w ith a liftin g to an a u to m o rp h is m a* of

F / S

such th a t

( M/ S ) a * = N / S .

P ro o f: If

F / M

a n d

F / N

are iso m o rp h ic, th e n th e re is an iso m o rp h ism

ip

fro m

F / M

to

F/ N.

It indu ces an a u to m o rp h is m

a

on

F / R,

b eca u se

(R/M)<p = R/ N.

B y th e le m m a above,

ip

h as a liftin g

ip*

to an a u to m o rp h is m of

F / S

b ecau se

F / S

is H opfian a n d

N / S

is finite. C learly,

ip*

in d uces

a

on

F / R

a n d m a p s

M / S

to

N/ S .

If th e re is an a u to m o rp h ism of

F / R

th a t lifts to an a u to m o rp h is m

a*

of

F / S

m a p p in g

M / S

to

N/ S ,

th e n

a*

in d u ces an iso m o rp h ism fro m

F / M

to

F/ N.

u

N ote th a t th e co n d itio n s of th e th e o re m do n o t req u ire

R

to be a c h a ra c te ristic s u b g ro u p of

F/ S.

In fact, th is w ould be too re s tric tiv e for p ra c tic a l p u rp o se s. Also n o te , th a t th e co n d itio n th a t, for all

M , N

£

Af

,

R / M

is m a p p e d to

R / N

u n d e r any iso m o rp h ism fro m

F / M

to

F / N

is a stro n g e r c o n d itio n th a n th e co n d itio n th a t

R / M

is c h a ra c te ristic in

F / M

for all

M

£ A

f.

If

Af

is a u n io n of c h a ra c te ristic classes, th e n th e liftin g s of all a u to m o rp h ism s

of

F / R

to

F / S

act on

Af.

Tw o su b g ro u p s

M

a n d

N

are in th e sam e o rb it if

a n d only if

F / M

a n d

F / N

are iso m o rp h ic. T h is m e an s th a t we can classify q u o tie n t g roups of

F / S

m od ulo su b g ro u p s in

Af

by d e te rm in in g th e o rb its of all liftin g s of a u to m o rp h ism s of

F/ R.

T h is fact was u sed , for ex am p le, in th e p -g ro u p g e n e ra tio n a lg o rith m as follows:

6.6 Corollary A ssum e th a t

F / R

is a fin ite p -g ro u p . Take 5 to be

[R, F]RP

a n d ta k e

Af

to be th e set of all su b g ro u p s of

F,

co n ta in e d in

R

a n d co n tain in g 5, such th a t for each

M

£

Af

th e su b g ro u p

R / M

of

F / M

is th e la st n o n -triv ia l te rm in th e low er p -c e n tra l series of

F/ M.

T h e n all a u to m o rp h is m s of

F / R

can be lifted to a u to m o rp h ism s of

F/ S.

Tw o su b g ro u p s

M

a n d

N

lie in th e sam e o rb it of

Af

u n d e r th e ac tio n of all such liftin g s if a n d only if th e q u o tien t grou p s

F / M

a n d

F / N

are iso m o rp h ic.

A d e ta ile d d escrip tio n of th e p -g ro u p g e n e ra tio n a lg o rith m can be fo u n d in O ’B rien (1990). T h e elem ents of

Af

are precisely th e so-called allowable subgroups

In document Central extensions of polycyclic groups (Page 81-86)