Group Theory Conference in Honor of V.D.Mazurov

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Small ranks of central unit groups of integral group rings of alternating groups

R. Aleev

In [1,2], the conditions are determined for the ranks to be equal to $0$ and $1$ .

Theorem. Let $r_n$ be a rank of central unit groups of integral group rings of alternating group of degree $n$ . Then

$\ \ 1)$ for $n\leqslant38$ , we have

$n$ $r_n$ $n$ $r_n$ $n$ $r_n$ $n$ $r_n$

$1$ $0$ $2$ $0$ $3$ $0$ $4$ $0$

$5$ $1$ $6$ $1$ $7$ $0$ $8$ $0$

$9$ $0$ $10$ $1$ $11$ $1$ $12$ $0$

$13$ $1$ $14$ $3$ $15$ $3$ $16$ $1$

$17$ $1$ $18$ $4$ $19$ $5$ $20$ $2$

$21$ $1$ $22$ $5$ $23$ $7$ $24$ $4$

$25$ $1$ $26$ $5$ $27$ $12$ $28$ $9$

$29$ $3$ $30$ $6$ $31$ $14$ $32$ $13$

$33$ $6$ $34$ $7$ $35$ $20$ $36$ $23$

$37$ $11$ $38$ $10$

$\ \ 2)$ for $n\geqslant39$ , we have $r_n\geqslant11$ .

References

R. A. Ferraz, Simple components and central units in group rings, J. Alg., 279, N1 (2004), 191–203.
R. Zh. Aleev, A. V. Kargapolov, V. V. Sokolov, The ranks of central unit groups of integral group rings of alternating groups, J. Math. Sci., 164, N2 (2010), 163–167.