3 Abelian periodic groups 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 compact

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, periodic group, a finitely generated group, involution, Golod group. 1. Preface In 1959, S. P. Novikov had

## Abstract

We classify the maximal irreducible periodic subgroups of PGL(q, $$ \mathbb{F} $$ ), where $$ \mathbb{F} $$ is a field of positive characteristic p transcendental over its prime subfield, q = p is prime, and $$ \mathbb{F} $$ × has an element of order q. That is, we construct a list of irreducible subgroups G of GL(q, $$ \mathbb{F} $$ ) containing the centre $$ \mathbb{F} $$ ×1q of GL(q, $$ \mathbb{F} $$ ), such that G/$$ \mathbb{F} $$ ×1q is a maximal periodic subgroup of PGL(q, $$ \mathbb{F} $$ ), and if H is another group of this kind then H is GL(q, $$ \mathbb{F} $$ )-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, $$ \mathbb{F} $$ ) is self-normalising.

Discrete Math. Appl., Vol. 12, No.5, pp. 459-475 (2002) © VSP2002. The structure of an infinite Sylow subgroup in some periodic Shunkov groups V. I. SENASHOV Abstract - We study periodic groups such that the normaliser of any finite non-trivial subgroup of such a group is almost layer-finite. The class of groups satisfying this condition is rather wide and includes the free Burnside groups of odd period which is greater than 665 and the groups constructed by A. Yu. Olshanskii. We consider the classical question: how the properties of the system of

J. Group Theory 17 (2014), 947–955 DOI 10.1515/ jgt-2014-0007 © de Gruyter 2014 Some groups of exponent 72 Enrico Jabara, Daria V. Lytkina and Victor D. Mazurov Communicated by Evgenii I. Khukhro Abstract. Local finiteness is proved for groups of exponent dividing 72 with no elements of order 6. 1 Introduction Our goal is to find conditions which guarantee local finiteness of a periodic group containing elements of orders 3 and 4 and no elements of order 6. The structure of finite groups with the same condition was described in [1]. Suppose that G is a periodic

Groups , Van Nostrand Mathematical Studies 16 , Van Nostrand Reinhold Co., New York–Toronto–London–Melbourne, 1969. [10] D.H. Hyers, On the stability of the linear functional equation , Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [11] S.V. Ivanov, The free Burnside groups of sufficiently large exponents , Internat. J. Algebra Comput. 4 (1994), no. 1–2, 1–308. [12] W. Magnus, A. Karrass, D. Solitar, Combinatorial Group Theory , Dover Publications, Inc., New York, 1976. [13] P.S. Novikov, S.I. Adian, Infinite periodic groups. 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 Book Review 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-periodic groups of motions in R 3 ) are known since 1919 a detailed tabulation has

Zeitschrift für Kristallographie 187, 319 (1989) © by R. Oldenbourg Verlag, München 1989 - 0044-2968/89 $3.00 + 0.00 Book Review 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-periodic groups of motions in R3) are known since 1919 a detailed tabulation has not been

Forum Math. 15 (2003), 665–677 Forum Mathematicum ( 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 periodic group G each sub- group has finite index in a permutable subgroup if and only if G