Classifying complements for groups. Applications
Annales de l'Institut Fourier, Volume 65 (2015) no. 3, p. 1349-1365

Let $A\le G$ be a subgroup of a group $G$. An $A$–complement of $G$ is a subgroup $H$ of $G$ such that $G=AH$ and $A\cap H=\left\{1\right\}$. The classifying complements problem asks for the description and classification of all $A$–complements of $G$. We shall give the answer to this problem in three steps. Let $H$ be a given $A$–complement of $G$ and $\left(▹,◃\right)$ the canonical left/right actions associated to the factorization $G=AH$. First, $H$ is deformed to a new $A$–complement of $G$, denoted by ${H}_{r}$, using a deformation map $r:H\to A$ of the matched pair $\left(A,H,▹,◃\right)$. Then the description of all complements is given: $ℍ$ is an $A$–complement of $G$ if and only if $ℍ$ is isomorphic to ${H}_{r}$, for some deformation map $r:H\to A$. Finally, the classification of complements proves that there exists a bijection between the isomorphism classes of all $A$–complements of $G$ and a cohomological object $𝒟\phantom{\rule{0.166667em}{0ex}}\left(H,A\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}\left(▹,◃\right)\right)$. As an application we show that the theoretical formula for computing the number of isomorphism types of all groups of order $n$ arises only from the factorization ${S}_{n}={S}_{n-1}{C}_{n}$.

Soit $G$ un groupe et $A\le G$ un sous-groupe de $G$. Un $A$–complément de $G$ est un sous-groupe $H$ de $G$ tel que $G=AH$ et $A\cap H=\left\{1\right\}$. Le problème auquel on s’intéresse est de classifier et décrire tous les $A$–compléments de $G$. Nous donnons la réponse à ce problème en trois étapes. Fixons $H$ un $A$–complément de $G$ et soient $\left(▹,◃\right)$ les actions canoniques associées à la factorisation $G=AH$. On commence par déformer $H$ en un nouveau $A$–complément ${H}_{r}$ à l’aide d’une certaine fonction $r:H\to A$ appelée fonction de déformation de $\left(A,H,▹,◃\right)$. Ensuite on donne la description de tous les $A$–compléments : $ℍ\le G$ est un $A$–complément de $G$ si et seulement si $ℍ$ est isomorphe à ${H}_{r}$ pour une certaine fonction de déformation $r:H\to A$. Enfin, la classification des $A$–compléments prouve qu’il existe une bijection entre les classes d’isomorphisme de tous les $A$–compléments de $G$ et un objet cohomologique $𝒟\phantom{\rule{0.166667em}{0ex}}\left(H,A\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}\left(▹,◃\right)\right)$. Comme application, on démontre que la formule qui calcule le nombre de classes d’isomorphisme des groupes d’ordre $n$ peut être retrouvée à partir de la factorisation ${S}_{n}={S}_{n-1}{C}_{n}$.

DOI : https://doi.org/10.5802/aif.2958
Classification:  20B05,  20B35,  20D06,  20D40
Keywords: Matched pairs, bicrossed products, the classification of finite groups
@article{AIF_2015__65_3_1349_0,
author = {Agore, Ana-Loredana and Militaru, Gigel},
title = {Classifying complements for groups. Applications},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {65},
number = {3},
year = {2015},
pages = {1349-1365},
doi = {10.5802/aif.2958},
language = {en},
url = {http://www.numdam.org/item/AIF_2015__65_3_1349_0}
}
Agore, Ana-Loredana; Militaru, Gigel. Classifying complements for groups. Applications. Annales de l'Institut Fourier, Volume 65 (2015) no. 3, pp. 1349-1365. doi : 10.5802/aif.2958. http://www.numdam.org/item/AIF_2015__65_3_1349_0/

[1] Agore, A. L. Classifying complements for associative algebras, Linear Algebra Appl., Tome 446 (2014), pp. 345-355 | Article | MR 3163149 | Zbl 1297.16006

[2] Agore, A. L.; Militaru, G. Classifying complements for Hopf algebras and Lie algebras, J. Algebra, Tome 391 (2013), pp. 193-208 | Article | MR 3081628 | Zbl 1293.16026

[3] Agore, Ana; Militaru, Gigel Schreier type theorems for bicrossed products, Cent. Eur. J. Math., Tome 10 (2012) no. 2, pp. 722-739 | Article | MR 2886568 | Zbl 1271.20038

[4] Arad, Zvi; Fisman, Elsa On finite factorizable groups, J. Algebra, Tome 86 (1984) no. 2, pp. 522-548 | Article | MR 732264 | Zbl 0526.20014

[5] Božović, Vladimir; Pace, Nicola On group factorizations using free mappings, J. Algebra Appl., Tome 7 (2008) no. 5, pp. 647-662 | Article | MR 2459096 | Zbl 1188.20022

[6] Douglas, Jesse On finite groups with two independent generators. I, II, III, IV, Proc. Nat. Acad. Sci. U. S. A., Tome 37 (1951), p. 604-610, 677–691, 749–760, 808–813 | Article | MR 45716 | Zbl 0043.02403 | Zbl 0044.01403 | Zbl 0044.01401

[7] Fisman, Elsa On the product of two finite solvable groups, J. Algebra, Tome 80 (1983) no. 2, pp. 517-536 | Article | MR 691811 | Zbl 0503.20005

[8] Fisman, Elsa; Arad, Zvi A proof of Szep’s conjecture on nonsimplicity of certain finite groups, J. Algebra, Tome 108 (1987) no. 2, pp. 340-354 | Article | MR 892909 | Zbl 0614.20013

[9] Gentchev, Ts. R. Factorizations of the sporadic simple groups, Arch. Math. (Basel), Tome 47 (1986) no. 2, pp. 97-102 | Article | MR 859256 | Zbl 0578.20008

[10] Giudici, Michael Factorisations of sporadic simple groups, J. Algebra, Tome 304 (2006) no. 1, pp. 311-323 | Article | MR 2256393 | Zbl 1107.20019

[11] Gorenstein, Daniel; Herstein, I. N. On the structure of certain factorizable groups. II, Proc. Amer. Math. Soc., Tome 11 (1960), pp. 214-219 | Article | MR 111786 | Zbl 0216.08203

[12] Hajós, Georg Über einfache und mehrfache Bedeckung des $n$-dimensionalen Raumes mit einem Würfelgitter, Math. Z., Tome 47 (1941), pp. 427-467 | Article | MR 6425 | Zbl 0025.25401

[13] Itô, Noboru Über das Produkt von zwei abelschen Gruppen, Math. Z., Tome 62 (1955), p. 400-401 | Article | MR 71426 | Zbl 0064.25203

[14] Liebeck, Martin W.; Praeger, Cheryl E.; Saxl, Jan The maximal factorizations of the finite simple groups and their automorphism groups, Mem. Amer. Math. Soc., Tome 86 (1990) no. 432, pp. iv+151 | Article | MR 1016353

[15] Liebeck, Martin W.; Praeger, Cheryl E.; Saxl, Jan On factorizations of almost simple groups, J. Algebra, Tome 185 (1996) no. 2, pp. 409-419 | Article | MR 1417379 | Zbl 0862.20016

[16] Liebeck, Martin W.; Praeger, Cheryl E.; Saxl, Jan Regular subgroups of primitive permutation groups, Mem. Amer. Math. Soc., Tome 203 (2010) no. 952, pp. vi+74 | Article | MR 2588738 | Zbl 1198.20002

[17] Maillet, Ed. Sur les groupes échangeables et les groupes décomposables, Bull. Soc. Math. France, Tome 28 (1900), pp. 7-16 | MR 1504357

[18] Praeger, Cheryl E.; Schneider, Csaba Factorisations of characteristically simple groups, J. Algebra, Tome 255 (2002) no. 1, pp. 198-220 | Article | MR 1935043 | Zbl 1014.20012

[19] Rotman, Joseph J. An introduction to the theory of groups, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 148 (1995), pp. xvi+513 | Article | MR 1307623 | Zbl 0810.20001

[20] Szép, J. Über die als Produkt zweier Untergruppen darstellbaren endlichen Gruppen, Comment. Math. Helv., Tome 22 (1949), p. 31-33 (1948) | Article | MR 26654 | Zbl 0036.15303

[21] Szép, J. Zur Theorie der endlichen einfachen Gruppen, Acta Sci. Math. Szeged, Tome 14 (1951), p. 111-112 | MR 48439 | Zbl 0043.25902

[22] Szép, J.; Rédei, L. On factorisable groups, Acta Univ. Szeged. Sect. Sci. Math., Tome 13 (1950), pp. 235-238 | MR 48433 | Zbl 0039.25502

[23] Takeuchi, Mitsuhiro Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra, Tome 9 (1981) no. 8, pp. 841-882 | Article | MR 611561 | Zbl 0456.16011

[24] Walls, Gary L. Groups which are products of finite simple groups, Arch. Math. (Basel), Tome 50 (1988) no. 1, pp. 1-4 | Article | MR 925486 | Zbl 0611.20017

[25] Wiegold, James; Williamson, Alan G. The factorisation of the alternating and symmetric groups, Math. Z., Tome 175 (1980) no. 2, pp. 171-179 | Article | MR 597089 | Zbl 0424.20004