Products of Conjugacy Classes in Groups
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1112 Products of Conjugacy Classes in Groups
Edited by Z. Arad and M. Herzog
Springer-Verlag Berlin Heidelberg New York Tokyo
Editors
Zvi Arad Department of Mathematics, Bar-han University Rarnat-Gan, Israel Marcel Herzog School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel
AMS Subject Classification (1980): 20-02, 20-04, 20A05, 20A15, 20B30, 20B99, 20C05, 20C07, 20C15, 20C30, 20C32, 20D05, 20D06, 20D08 ISBN 3-540-13916-8 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13916-8 Springer-Verlag New York Heidelberg Berlin Tokyo
This work is subject to copyright. All rights are reserved, whether the whole or part ofthe material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach /Bergstr. 2146/3140-543210
Dedicated to the memory of
DR. RITA HERZOG the late wife of the second editor
Products of Conjugacy Classes in Groups
CONTENTS Introduction,
Chapter 1,
Powers and products of conjugacy classes in groups by Z, Arad, M. Herzog and J. Stavi
Chapter 2.
•••••.
Covering numbers of groups of small order and sporadlc groups by S. Ka rn l (under the supervision of Z. Arad]
Chapte r 3,
52
Covering properties of permutation groups by y, Dvir (under the supervision of Z, Arad)
Chapter 4,
6
197
Groups with a small covering number by Z, Arad, D, Chillag and G. Moran
222
Introduction. This book presents recent progress on covering theorems for simple groups.
These results, as well as the methods, have applicat-
ions in diverse areas of group theory. Let
G
be a group,
C
any nontrivial conjugacy class of
G.
We define a regular covering theorem to be a theorem which guarantees, under certain conditions, that for a positive integer m and every conjugacy class
C
m C = G.
1,
An extended covering theorem
guarantees, under appropriate conditions, that for a positive integer r
r,
C
IT
i=l
i
= G for every sequence
C
l'C 2""'C r
distinct) nontrivial conjugacy classes of
of (not necessarily
G.
The basic theorems establish the existence of such
m
and
r
for finite nonabelian simple groups and for certain types of infinite simple groups.
While the general problem of determining minimal values
for
is as yet unsolved, we have obtained a number of
m
and
r
interesting results. ecn(G) and
We denote by
cn(G)
(extended covering number of r,
respectively.
Clearly
G)
cn(G)
(covering number of G) the minimal values for
and m
ecn(G).
Covering theorems for finite simple groups were first studied by J. Brenner and his associates [2-10J. n
S, they determined conjugacy classes
where n
m = 2, 3, or 4.
6,
satisfying
C
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