On group-theoretic decision problems and their classification

120 Pages·1971·8.998 MB·English

by  Miller, C. F.

Annals of Mathematics Studies
Number 68

ON GROUP-THEORETIC
DECISION PROBLEMS
AND THEIR CLASSIFICATION
BY
CHARLES F. MILLER, III
PRINCETON UNIVERSITY PRESS
AND
UNIVERSITY OF TOKYO PRESS
PRINCETON, NEW JERSEY
1971
Copyright © 1971, by Princeton University Press
ALL RIGHTS RESERVED
LC Card: 70-146647
ISBN: 0-691-0-08091-7
AMS, 1968: 0275, 2010
Published in Japan exclusively by
University of Tokyo Press;
in other parts of the world by
Princeton University Press
Printed in the United States of America
PREFACE
This study is concerned with group-theoretic decision problems -
specifically the word problem, conjugacy problem, and isomorphism problem
for finitely presented groups. The principal aims of the work are twofold:
(1) to prove unsolvability results for decision problems in classes of finite
ly presented groups which are in some sense "elementary" (for example,
residually finite); and (2) to obtain some new unsolvability results for the
isomorphism problem. Of course, there is some overlap between these two
goals.
The main objective is to present these new results, but a variety of
background material has been included, and proofs of several background
results are supplied. Although not intended as a textbook, it is hoped that
inclusion of background will help make the monograph accessible to a
larger mathematical audience. A proof of the unsolvability of the word
problem has not been included, but this is now available in at least two
textbooks ([ 44], [ 48]).
The Introduction (Chapter I) contains formulations of the decision
problems considered, a brief survey of the field, and statements of the
principal new results. In Chapter II, several useful group-theoretic results
concerning equality and conjugacy of words in certain groups are con
sidered. Chapter III is concerned with unsolvability in certain "elemen
tary" groups. In Chapter IV, the more difficult arbitrary r.e. degree results
for "elementary" groups are proved, and then applied to the isomorphism
problem. Finally, Chapter V is devoted to obtaining a strong form of the
unsolvability of the isomorphism problem for finitely presented groups in
general.
This work had its origin in my doctoral dissertation [53], written at the
University of Illinois.
v
vi PREFACE
I wish to thank William W. Boone for his invaluable advice, guidance,
and encouragement during the preparation of this work. I would also like
to thank K. I. Appel, D. ] . Collins, and P. E. Schupp for their interest,
encouragement, and suggestions, and to thank ] . Mary Tyrer for her help
with the proofs. Partial support from the U. S. National Science Founda
tion is gratefully acknowledged.
Charles F. Miller, III
CONTENTS
PREFACE v
CHAPTER I
INTRODUCTION 1
CHAPTER II
PROPERTIES OF BRITTON EXTENSIONS
A. Britton's Theorem A .................................................................... 13
B. Collins' Lemma Generalized ...................................................... 20
C. Further Properties of Britton Extensions 24
CHAPTER III
UNSOLVABILITY RESULTS FOR RESIDUALLY FINITE GROUPS
A. A F .P. Residually Finite Group with Unsolvable
Conjugacy Problem ...................................................................... 2S
B. Unsolvable Problems in the Automorphism Group
of a Free Group ............................................................................ 31
C. Unsolvable Problems in Direct Products of
Free. Groups .... .............................. ........................................ ...... .. 3S
D. On a Problem of Graham Higman ................................................ 42
CHAPTER IV
THE WORD AND CONJUGACY PROBLEMS FOR CERTAIN
ELEMENT ARY GROUPS
A. A Group Theoretic Characterization of Turing
Reducibility .......................... ...... .................................................. 4S
B. Degree Results for the Conjugacy Problem in
Elementary Groups SS
C. Some Applications 73
vii
viii CONTENTS
CHAPTER V
ON THE ISOMORPHISM PROBLEM FOR GROUPS
A. Background and Results .............. ................................ ................ 79
B. The Basic Construction .................................................... .......... 80
C. Rabin Revisited ............ .................................... ............................ 88
D. Generalizations ............................................................................ 90
E. Isomorphism Problems of Arbitrary Recursively
Enumerable Many-one Degree ...................................... .............. 93
F. Supplement on Certain Commutators .......................................... 95
REFERENCES .............. ...... ...................................................................... 97
INDEX OF SYMBOLS ................................................................................ 103
INDEX ........................................................................................................ 105