Geodesic
Maximal subgroup growth of a~few polycyclic groups
Algebra and discrete mathematics, Tome 32 (2021) no. 2, pp. 226-235.

Voir la notice de l'article provenant de la source Math-Net.Ru

We give here the exact maximal subgroup growth of two classes of polycyclic groups. Let $G_k = \langle x_1, x_2, \dots , x_k \mid x_ix_jx_i^{-1}x_j \text{ for all } i j \rangle$, so $G_k = \mathbb{Z} \rtimes (\mathbb{Z} \rtimes (\mathbb{Z} \rtimes \dots \rtimes \mathbb{Z}))$. Then for all integers $k \geq 2$, we calculate $m_n(G_k)$, the number of maximal subgroups of $G_k$ of index $n$, exactly. Also, for infinitely many groups $H_k$ of the form $\mathbb{Z}^2 \rtimes G_2$, we calculate $m_n(H_k)$ exactly.
Keywords: maximal subgroup growth, polycyclic groups, semidirect products. 
@article{ADM_2021_32_2_a4,
     author = {A. Kelley and E. Wolfe},
     title = {Maximal subgroup growth of a~few polycyclic groups},
     journal = {Algebra and discrete mathematics},
     pages = {226--235},
     publisher = {mathdoc},
     volume = {32},
     number = {2},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2021_32_2_a4/}
}
TY  - JOUR
AU  - A. Kelley
AU  - E. Wolfe
TI  - Maximal subgroup growth of a~few polycyclic groups
JO  - Algebra and discrete mathematics
PY  - 2021
SP  - 226
EP  - 235
VL  - 32
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ADM_2021_32_2_a4/
LA  - en
ID  - ADM_2021_32_2_a4
ER  - 
%0 Journal Article
%A A. Kelley
%A E. Wolfe
%T Maximal subgroup growth of a~few polycyclic groups
%J Algebra and discrete mathematics
%D 2021
%P 226-235
%V 32
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ADM_2021_32_2_a4/
%G en
%F ADM_2021_32_2_a4
A. Kelley; E. Wolfe. Maximal subgroup growth of a~few polycyclic groups. Algebra and discrete mathematics, Tome 32 (2021) no. 2, pp. 226-235. http://geodesic.mathdoc.fr/item/ADM_2021_32_2_a4/

[1] K. Brown, Cohomology of groups, Springer-Verlag, New York, 1982 | MR | Zbl

[2] E. Gelman, “Subgroup growth of Baumslag-Solitar groups”, J. Group Theory, 8:6 (2005), 801–806 | DOI | MR | Zbl

[3] A. Jaikin-Zapirain and L. Pyber, “Random generation of finite and profinite groups and group enumeration”, Ann. of Math. (2), 173:2 (2011), 769–814 | DOI | MR | Zbl

[4] A. Kelley, Maximal Subgroup Growth of Some Groups, Ph.D. thesis, State University of New York at Binghamton, 2017 | MR

[5] A. Kelley, “Subgroup growth of all Baumslag-Solitar groups”, New York J. Math., 2020, 218–229 ; http://nyjm.albany.edu/j/2020/26-11.html | MR | Zbl

[6] A. Kelley, Maximal subgroup growth of some metabelian groups, to appear in Comm. Algebra, preprint, 2018, arXiv: 1807.03423 | MR

[7] A. Lubotzky and D. Segal, Subgroup growth, Birkhauser Verlag, Basel, 2003 | MR | Zbl

[8] D. Robinson., A course in the theory of groups, 2nd edition, Springer-Verlag, New York, 1996 | MR

[9] A. Shalev, “On the degree of groups of polynomial subgroup growth”, Trans. Amer. Math. Soc., 351:9 (1999), 3793–3822 | DOI | MR | Zbl