Zero divisors of support size 3 in group algebras and trinomials divided by irreducible polynomials over GF(2)

  • Alireza Abdollahi

    University of Isfahan, Iran; and Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
  • Zahra Taheri

    University of Isfahan, Iran
Zero divisors of support size 3 in group algebras and trinomials divided by irreducible polynomials over GF(2) cover
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Abstract

A famous conjecture about group algebras of torsion-free groups states that there is no zero divisor in such group algebras. A recent approach to settle the conjecture is to show the non-existence of zero divisors with respect to the length of possible ones, where by the length we mean the size of the support of an element of the group algebra. The case length 2 cannot be happen. The first unsettled case is the existence of zero divisors of length 3. Here we study possible length 3 zero divisors in the rational group algebras and in the group algebras over the field Fp\mathbb{F}_p with pp elements for some prime pp. As a consequence we prove that the rational group algebras of torsion-free groups which are residually finite pp-group for some prime p3p\neq 3 have no zero divisor of length 3. We note that the determination of all zero divisors of length 3 in group algebras over F2\mathbb{F}_2 of cyclic groups is equivalent to find all trinomials (polynomials with 3 non-zero terms) divided by irreducible polynomials over F2\mathbb{F}_2. The latter is a subject studied in coding theory and we add here some results, e.g. we show that 1+x+x21+x+x^2 is a zero divisor in the group algebra over F2\mathbb{F}_2 for some element xx of the group if and only if xx is of finite order divided by 33 and we find all β\beta in the group algebra of the shortest length such that (1+x+x2)β=0(1+x+x^2)\beta=0; and 1+x2+x31+x^2+x^3 or 1+x+x31+x+x^3 is a zero divisor in the group algebra over F2\mathbb{F}_2 for some element xx of the group if and only if xx is of finite order divided by 7.

Cite this article

Alireza Abdollahi, Zahra Taheri, Zero divisors of support size 3 in group algebras and trinomials divided by irreducible polynomials over GF(2). Rend. Sem. Mat. Univ. Padova 145 (2021), pp. 191–203

DOI 10.4171/RSMUP/78