3 Abelian periodicgroups
In this chapter we shall exclusively treat the class of topological abelian groups that
have been called periodic.
Notation 3.1. Every natural number n has, for a given prime p, a presentation
n = pνn,
where p does not divide n. We let νp(n) denote this number ν.
3.1 Braconnier’s theorem
Let us recall from Definition 1.13 that a locally compact abelian group G is periodic
provided it is totally disconnected and is a union of compact subgroups.
Since a totally disconnected locally compact abelian group has arbitrarily small
construction is generalized. We give a condition that guarantees that finitely
generated subalgebra of nilalgebra is not nilpotent. The infinite subgroups of the Golod
group generated by involutions are constructed.
The work was supported by the Ministry of Education grant, the theme No. 1.34.11 and
by the Krasnoyarsk State Pedagogical University V. P. Astafieva, grant NSH No. 10.
Keywords: free associative algebra, homogeneous ideal nilalgebra, nilpotent algebra,
periodicgroup, a finitely generated group, involution, Golod group.
In 1959, S. P. Novikov had
We classify the maximal irreducible periodic subgroups of PGL(q, $$
), where $$
is a field of positive characteristic p transcendental over its prime subfield, q = p is prime, and $$
× has an element of order q. That is, we construct a list of irreducible subgroups G of GL(q, $$
) containing the centre $$
×1q of GL(q, $$
), such that G/$$
×1q is a maximal periodic subgroup of PGL(q, $$
), and if H is another group of this kind then H is GL(q, $$
)-conjugate to a group in the list. We give criteria for determining when two listed groups are conjugate, and show that a maximal irreducible periodic subgroup of PGL(q, $$
) is self-normalising.
Groups , Van Nostrand Mathematical Studies 16 , Van Nostrand Reinhold Co., New York–Toronto–London–Melbourne, 1969.  D.H. Hyers, On the stability of the linear functional equation , Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224.  S.V. Ivanov, The free Burnside groups of sufficiently large exponents , Internat. J. Algebra Comput. 4 (1994), no. 1–2, 1–308.  W. Magnus, A. Karrass, D. Solitar, Combinatorial Group Theory , Dover Publications, Inc., New York, 1976.  P.S. Novikov, S.I. Adian, Infinite periodicgroups. I–III. (Russian) , Izv. Akad
Zeitschrift für Kristallographie 187. 319 (1989)
c by R. Oldenbourg Verlag. München 1989 - 0044-2968/89 $3.00 + 0.00
Tables of the 80 plane groups in three dimensions
By Hildegard Grell, Christa Krause and Juliane Grell
iir. Inform. Inf. Rep., Sonderausgabe No. 2,
edited by the Institut für Informatik und Rechentechnik
der Akademie der Wissenschaften der DDR, Rudower Chaussee 5, Berlin, 1199.
Pp. 119. Price about DM 80 (paperback).
Although the layer groups (2-periodicgroups of motions in R 3 ) are known since 1919 a
detailed tabulation has
Forum Math. 15 (2003), 665–677 Forum
( de Gruyter 2003
Groups in which every subgroup is nearly permutable
M. De Falco, F. de Giovanni, C. Musella, Y. P. Sysak1
(Communicated by Rüdiger Göbel)
Abstract. A relevant theorem of B. H. Neumann states that in a group G each subgroup has
finite index in its normal closure if and only if the commutator subgroup G 0 of G is finite, i.e. if
and only if G is finite-by-abelian. In this article we prove that in a periodicgroup G each sub-
group has finite index in a permutable subgroup if and only if G