Explicit free groups in division rings

Gonçalves, J.; Passman, D.

doi:10.1090/S0002-9939-2014-12230-1

Explicit free groups in division rings
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by J. Z. Gonçalves and D. S. Passman

Proc. Amer. Math. Soc. 143 (2015), 459-468

DOI: https://doi.org/10.1090/S0002-9939-2014-12230-1

Published electronically: October 1, 2014

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Abstract:

Let $D$ be a division ring of characteristic $\neq 2$ and suppose that the multiplicative group $D^\bullet =D\setminus \{0\}$ has a subgroup $G$ isomorphic to the Heisenberg group. Then we use the generators of $G$ to construct an explicit noncyclic free subgroup of $D^\bullet$. The main difficulty occurs here when $D$ has characteristic $0$ and the commutators in $G$ are algebraic over $\mathbb {Q}$.

References

Jairo Z. Gonçalves, Arnaldo Mandel, and Mazi Shirvani, Free products of units in algebras. I. Quaternion algebras, J. Algebra 214 (1999), no. 1, 301–316. MR 1684864, DOI 10.1006/jabr.1998.7680
Jairo Z. Gonçalves, Arnaldo Mandel, and Mazi Shirvani, Free products of units in algebras. II. Crossed products, J. Algebra 233 (2000), no. 2, 567–593. MR 1793917, DOI 10.1006/jabr.2000.8440
J. Z. Gonçalves and M. Shirvani, A survey on free objects in division rings and in division rings with an involution, Comm. Algebra 40 (2012), no. 5, 1704–1723. MR 2924478, DOI 10.1080/00927872.2011.554934
Gerald J. Janusz, Algebraic number fields, 2nd ed., Graduate Studies in Mathematics, vol. 7, American Mathematical Society, Providence, RI, 1996. MR 1362545, DOI 10.1090/gsm/007
A. Lichtman, On subgroups of the multiplicative group of skew fields, Proc. Amer. Math. Soc. 63 (1977), no. 1, 15–16. MR 447432, DOI 10.1090/S0002-9939-1977-0447432-0
A. I. Lichtman, Free subgroups of normal subgroups of the multiplicative group of skew fields, Proc. Amer. Math. Soc. 71 (1978), no. 2, 174–178. MR 480623, DOI 10.1090/S0002-9939-1978-0480623-2
A. I. Lichtman, On normal subgroups of multiplicative group of skew fields generated by a polycyclic-by-finite group, J. Algebra 78 (1982), no. 2, 548–577. MR 680374, DOI 10.1016/0021-8693(82)90095-3
Ian Stewart and David Tall, Algebraic number theory and Fermat’s last theorem, 3rd ed., A K Peters, Ltd., Natick, MA, 2002. MR 1876804
A. E. Zalesskiĭ, The group algebras of solvable groups, Vescī Akad. Navuk BSSR Ser. Fīz.-Mat. Navuk 1970 (1970), no. 2, 13–21 (Russian). MR 0289675