Geodesic
On finite Alperin $p$-groups with homocyclic commutator subgroup
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 17 (2011) no. 4, pp. 53-65
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We study metabelian Alperin groups, i.e., metabelian groups in which every 2-generated subgroup has a cyclic commutator subgroup. It is known that, if the minimum number of generators $d(G)$ of a finite Alperin $p$-group $G$ is $n\geq3$, then $d(G')\leq C_n^2$ for $p\neq3$ and $d(G')\leq C_n^2+C_n^3$ for $p=3$. The first section of the paper deals with finite Alperin $p$-groups $G$ with $d(G)\geq3$ and $p\neq3$ that have a homocyclic commutator subgroup of rank $C_n^2$. In addition, a corollary is deduced for infinite Alperin $p$-groups. In the second section, we prove that, if $G$ is a finite Alperin $3$-group with a homocyclic commutator subgroup $G'$ of rank $C_n^2+C_n^3$, then $G'$ is an elementary abelian group.
Mots-clés : $p$-group
Keywords: Alperin group, commutator subgroup, definition of group by means of generators and defining relations. 
@article{TIMM_2011_17_4_a4,
     author = {B. M. Veretennikov},
     title = {On finite {Alperin} $p$-groups with homocyclic commutator subgroup},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {53--65},
     year = {2011},
     volume = {17},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2011_17_4_a4/}
}
TY  - JOUR
AU  - B. M. Veretennikov
TI  - On finite Alperin $p$-groups with homocyclic commutator subgroup
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2011
SP  - 53
EP  - 65
VL  - 17
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TIMM_2011_17_4_a4/
LA  - ru
ID  - TIMM_2011_17_4_a4
ER  - 
%0 Journal Article
%A B. M. Veretennikov
%T On finite Alperin $p$-groups with homocyclic commutator subgroup
%J Trudy Instituta matematiki i mehaniki
%D 2011
%P 53-65
%V 17
%N 4
%U http://geodesic.mathdoc.fr/item/TIMM_2011_17_4_a4/
%G ru
%F TIMM_2011_17_4_a4
B. M. Veretennikov. On finite Alperin $p$-groups with homocyclic commutator subgroup. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 17 (2011) no. 4, pp. 53-65. http://geodesic.mathdoc.fr/item/TIMM_2011_17_4_a4/

[1] Alperin J. L., “On a special class of regular groups”, Trans. Amer. Math. Soc., 106:1 (1963), 77–99 | MR | Zbl

[2] Veretennikov B. M., “Ob odnoi gipoteze Alperina”, Sib. mat. zhurn., 21:1 (1980), 200–202 | MR | Zbl

[3] Veretennikov B. M., “O konechnykh 3-porozhdennykh 2-gruppakh Alperina”, Sib. elektron. mat. izv., 4 (2007), 155–168 | MR | Zbl

[4] Veretennikov B. M., “Konechnaya 2-gruppa Alperina s vtorym kommutantom proizvolnogo poryadka”, Maltsevskie chteniya, tez. dokl. Mezhdunar. konf., Novosibirsk, 2009, 47

[5] Veretennikov B. M., “O range vtorykh kommutantov 2-grupp Alperina”, Maltsevskie chteniya, tez. dokl. Mezhdunar. konf., Novosibirsk, 2010, 69

[6] Veretennikov B. M., “2-gruppy Alperina s beskonechnymi tsiklicheskimi i elementarnymi abelevymi vtorymi kommutantami”, Algebra, logika, teoriya i prilozheniya, tez. dokl. Mezhdunar. konf., Krasnoyarsk, 2009, 15

[7] Huppert B., Endliche Gruppen, v. I, Springer-Verlag, Berlin, 1967, 796 pp. | MR | Zbl

[8] Veretennikov B. M., “O konechnykh 2-gruppakh Alperina s tsiklicheskimi vtorymi kommutantami”, Algebra i logika, 50:3 (2011), 326–350

[9] Kholl M., Teoriya grupp, IL, M., 1962, 468 pp.