Geodesic
The ranks of central unit groups of integral group rings of alternating groups
Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 7, pp. 15-21 
R. Zh. Aleev; A. V. Kargapolov; V. V. Sokolov. The ranks of central unit groups of integral group rings of alternating groups. Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 7, pp. 15-21. http://geodesic.mathdoc.fr/item/FPM_2008_14_7_a1/
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     author = {R. Zh. Aleev and A. V. Kargapolov and V. V. Sokolov},
     title = {The ranks of central unit groups of integral group rings of alternating groups},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2008_14_7_a1/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $G$ be a finite group and $\mathrm U(Z(\mathbf ZG))$ be the group of units of the center $Z(\mathbf ZG)$ of the integral group ring $\mathbf ZG$ (the central unit group of the ring $\mathbf ZG$). The purpose of the present work is to study the ranks $r_n$ of groups $\mathrm U(Z(\mathbf Z\mathrm A_n)$, i.e., of central unit groups of integral group rings of alternating groups $\mathrm A_n$. We shall find all values $n$ for $r_n=1$ and propose an approach how to describe the groups $\mathrm U(Z(\mathbf Z\mathrm A_n))$ in these cases, and we will present some results of calculations of $r_n$ for $n\leq600$.

[1] Aleev R. Zh., “Edinitsy polei kharakterov i tsentralnye edinitsy tselochislennykh gruppovykh kolets konechnykh grupp”, Mat. trudy, 3:1 (2000), 3–37 | MR | Zbl

[2] Frobenius G., Teoriya kharakterov i predstavlenii grupp., Kharkovskaya mat. biblioteka. Kniga tretya, Gos. nauch.-tekhn. izd. Ukrainy, Kharkov, 1937

[3] Endryus G., Teoriya razbienii, Nauka, M., 1982 | MR

[4] Aleev R. Ž., “Higman's central unit theory, units of integral group rings of finite cyclic groups and Fibonacci numbers”, Internat. J. Algebra Comput., 4:3 (1994), 309–358 | DOI | MR | Zbl

[5] Arora S. R., Hales A. W., Passi I. B. S., “Jordan decomposition and hypercentral units in integral group ring”, Commun. Algebra, 21:1 (1993), 25–35 | DOI | MR | Zbl

[6] Arora S. R., Passi I. B. S., “Central height of the unit group of integral group ring”, Commun. Algebra, 21:10 (1993), 3673–3683 | DOI | MR | Zbl

[7] Ferraz R. A., “Simple components and central units in group rings”, J. Algebra, 279:1 (2004), 191–203 | DOI | MR | Zbl

[8] Giambruno A., Jespers E., “Central idempotents and units in rational group algebras of alternating groups”, Internat. J. Algebra Comput., 8:4 (1998), 467–477 | DOI | MR | Zbl

[9] The GAP Group, GAP Groups, Algorithms, and Programming, Version 4.4.2 , 2004 http://www.gap-system.org