This thesis examines automorphisms of certain relatively free groups and one relator groups. In Chapter 3, the author derives an explicit criterion to determine whether certain endomorphisms of free groups of finite rank are automorphisms. Non-tame automorphisms are also constructed. Chapter 4 shows that for certain two generator torsion-free one relator groups G, the automorphism group Aut(G) coincides with the group of inner automorphisms. Chapter 5 obtains a characterization of semi-complete groups analogous to Baer's direct factor property criterion for complete groups. Several examples are provided to demonstrate situations where the group of inner automorphisms equals the full automorphism group.

1. AUTOMORPHISMS OF CE豊TAIN 豊電LATI VELY
FREE GROUPS AND ON電 R鴬LATOR GROUPS
By
N. CHANDFIAMOWLISWARAN
Department of Mathematics
Thisな submit加d
加 fuif/ment of requirements for
the award of the degree of
DOCTOR OF PHILOSOPHY
INDIAN INSTITUTE OF TECHNOLOGY, DELHI
INDiA
NOVEMBER 1996

2. ず、H
ろ)需・5呂観・島
乙有月，A
ご＿A

3. J7.1 71Lj b(iO'VCd 1) (I r・f n is
v . Nar司ranaswamy
N. i'iatliuraxn

4. Certi丑cate
-Fhis is to certify that tule thesis entitled "Automorphisrns of Certain
Relatively Free Groups and One Relator Groups" whlich is being
SUblflit,ted by N. Chandramowliswaran 1r the award of DOCTOR
OF PHILOSOPHY (MATHEMATICS)to thle Indian Institute of
Technology, Delhi, is a record of bonafide researchl 'vork carried ont by
liiin unlder・mlly gui danlce ari (1 S tiper・vi s i oli.Tfie t}ic s is has reachec,l the
standards fullfilling th（う reくluir・errients of the regulations relal・ing to t hc
degree. TFie i'esiili・r; obtained in the i,1iesis hlave not beenl subiliitllecl to
aity other Universit,y or I丁istitule 伽1・もhe award of i i.Ilう『 degree or diplollla.
喫醜示～帆ぬ一
J .B. Sriぬstava 熱劃・＼り
P iofes sor
rc1つartrnent of 八I at ii cui at i CS
1ii(117iIl Iiisl』itiilJ(! of F(: i:1iiio1ogy
Ihしし'z Iく」ins, N(:w t)elIii-11()fl II) L
I ii (lia.
,＼的ぐ
,
(;

5. ACKN OWLED GEMENT S
I am greatly indebted to my supervisor Professor J.B. Srivastava for
introducing me to Combinatorial Group Theory. His valuable guidance arid
excellent teaching has beneffited me in building up the background for this
research work. Bis tremendous patience and constant encouragement has
been a source of motivation for me.
i take this opportunity to thank Professor S・R・K. lyengar, Head,
Department of M戚hem航ics, for his encour昭ement and moral support in
completing this thesis. The kind help and association extended by the faculty
of Department of M誠，hem試ics, especially Professor N.S. Kambo and
Professor B.R. Janda is gr誠efully acknowledged. It is 叫 pie闘ure to
thank 町 friends V. Ravindranath, T.V. Selva Kumaran, K. Srinivasaii,M.
Arulanandam, R.K. Dash, Necia Nataraj,Soma Gupta a,nd all others for
keeping me in good spirits and cheers.
This research work is supported by the Council of Scientiffic and In-
dustrial Research and I owe immense gr誠・itude to CSIR, for providing
丘nancial support.
I owe so much to my sisters Saradha, Viji，ぬtchala and my brother N.
罷nkatesan for their e肥r loving wishes.
店
N. Chandramowliswaran

6. Notations
C Proper inclusion
9 Inclusion
_i/v Set of natural numbers
2 Set of integers
H 5 G .g is a subgrcup of G
N ョ Y . N is a normal subgroup of C
z（の Center of G
ュy J_1xy
[x, y」 ュー‘y-1x y
[z1，ユ 2,… ，xn,xn+1j [[xi，エ2, …，エni，ュ n+1 j
争、( G) The n th term of the lower centra,l series of G
島 (G) the (n+l)th term of the derived series of G
[ H, K] The subgroup generated by all commutators
[h,k]。 h e H, k E K
G' The commutator subgroup of G
j;; The free group on X:，及，…，ル
F The free group on Xi，義，…，瓦，，・・
ZC The integral Group ring of G
2凡 The integral Group ring of the free group P'?,
ハ凡 The augm帥t肌ion ideal of Z FL
GL, (2凡） The general linear group of degree n over 2凡
ムR The left ideal ofZ凡 generated by all e)emcnts
of the form {r 一 i s r E R, RS 凡｝
OL,-. (R) The general linear group of degree n over the ring R
OJJ。(2) rnI、e group of n x n matrices with integra.l entries
whose determinant is 士」

7. 111111 IllX
GGJA(G)
End（の
響－奴四
響＝改四
Automorphism group of G
The group of inner automorphisms of C
The kernel of the natural homomorphism
{Aut (G) → Aut (G/G')}
The semi-group of all endomorphisms of G
The left Fox derivative of the 丘eely reduced
word W E 凡 w.r.t 乃
The right Fox derivative of the freely reduced
word W E 凡 w.r.t 毛

8. ABSTRACT
As the title of the thesis "Autornorphisms of. Certain Relatively Free Groups arid Ciie
Relator Croups" suggests, our main aim in this thesis is to attempt and solve some problems
ori a.utomorphism groups of some relatively free groups and one relator groups of special
type. In chapter 1,wehave discussed the lower centra,l series, the derived series, residually
finite groups, varieties of groups and relatively free groups. i施 have given a brief and up-to-
date survey of known results relevant to our works in this thesis together with our comment.s
and remarks. In chapter 2, we have obtained some results ori group rings and applied
free di無rential calculus, developed by 恥X to get a new t田t word for determining cei・tahi
automorphisms. We also discussed J.S. Birman's theorem and Krasnikov's theorem to get a
relation between automorphisms arid invertible matrices over certain integral group rings. [n
chapter 3,we h加obtained several important results ofthe thesis related to autoinorphisms
of cci・tain relatively fr肥 groups. It contains several results which play a signifficant role in
liれing automorphisrns and non・tame automorphisms of certain groups.
We hlave derived a very explicit criterion for determining whether certain special types of
eridomorpiijsms of free groups of fluite rank are automorphisms. We have also constructed
a class of non-tame automorpltisms of the groups F2/呪(R),m と 3, where .8 is a non ti・i "ial
normal subgroup of free group F2.
Chapter 4 deals with automorphisms of two generator one relator groups. For 叫i'tai n
two generator torsioii-free one relator group G, it has been shown that the autornorphism
group Aut（の of G coincides with the group of IA-automorphisms of G. Several examples ai・e
D
givenI・ln chapter り， we nave ootaincu a ciiaracしeri型凹四L semi compiete groups anaiogrnユs
to Baer's direct ねctor property criterion for complete groups. Many more results and severa.l
examples are given to demonstrate the situations where IA(G)=Jnrz(G). At the end we
have givenl some concludling remarks and recorded 9 problems which arose during our work.

9. i
12
23
加
29
43
53
61
61
74
Contents
i Introduction
i.i Preliminaries
2 Automorphisms of free groups, free solvable groups and
relatively free groups
i .3 A Brief Outline
2 Group Rings and Free Di卿rential Calculus
2.1 Group Rings ..... ・・
2.2 Free Di髭rential Calculus
2.3 Automorphisms of Relatively Free Groups
3 Automorphisms of Some Relatively Free Groups
3.1 Automorphisrns of Certain Relatively Free Groups
3'2 Determination of Certain JA-Automorphisms
3,3 Some Applications
81
踊
4 One Relator Groups

10. 4.2 Automorphisms of Certain Torsion-free T如－genera切r
One Relator Groups . . . ．。． . . . . . . .,. . ．、． . 90
4.3 Example . . . .. . . . . . . . . . .,. . . . . . . . .,. 99
5 Sorne More Results onAutomorphism Groups 105
5.1 Semicomplete Groups . . . . . . . . .'. . . . . . . . . . I 05
52 Some more Examples .,. . . . . . . . . . . .',. . . . 116
5.3 Concluding Remarks . . .,,．』‘ . . . . . . . . .,. . i 20