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Perhaps it is not inappropriate for me to begin with the comment
that this book has been an interesting challenge to the translator.
It is most unusual, in a text of this type, in that the style is
racy, with many literary allusions and witticisms: not the easiest
to translate, but a source of inspiration to continue through
material that could daunt by its combinatorial complexity.
Moreover, there have been many changes to the text during the
translating period, reflecting the ferment that the subject of the
restricted Burnside problem is passing through at present. I concur
with Professor Kostrikin's "Note in Proof', where he describes the
book as fortunate. I would put it slightly differently: its
appearance has surely been partly instrumental in inspiring much
endeavour, including such things as the paper of A. I. Adian and A.
A. Razborov producing the first published recursive upper bound for
the order of the universal finite group B(d,p) of prime exponent
(the English version contains a different treatment of this result,
due to E. I. Zel'manov); M. R. Vaughan-Lee's new approach to the
subject; and finally, the crowning achievement of Zel'manov in
establishing RBP for all prime-power exponents, thereby (via the
classification theorem for finite simple groups and Hall-Higman)
settling it for all exponents. The book is encyclopaedic in its
coverage of facts and problems on RBP, and will continue to have an
important influence in the area.
Three years have passed since the publication of the Russian
edition of this book, during which time the method described has
found new applications. In [26], the author has introduced the
concept of the periodic product of two groups. For any two groups G
and G without elements of order 2 and for any 1 2 odd n ~ 665, a
group G @ Gmay be constructed which possesses several in 1 2
teresting properties. In G @ G there are subgroups 6 and 6
isomorphic to 1 2 1 2 G and G respectively, such that 6 and 6
generate G @ G and intersect 1 2 1 2 1 2 in the identity. This
operation "@" is commutative, associative and satisfies Mal'cev's
postulate (see [27], p. 474), i.e., it has a certain hereditary
property for subgroups. For any element x which is not conjugate to
an element of either 6 1 or 6 , the relation xn = 1 holds in G @ G
* From this it follows that when 2 1 2 G and G are periodic groups
of exponent n, so is G @ G * In addition, if G 1 2 1 2 1 and G are
free periodic groups of exponent n the group G @ G is also free 2 1
2 periodic with rank equal to the sum of the ranks of G and G * I
believe that groups 1 2
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