Nilpotency class of symmetric units of group algebras

Zsolt Balogh, Tibor Juhász

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


Let F be a field of odd prime characteristic p, G a group, U the group of units in the group algebra FG, and U+ the subgroup of U generated by the elements of U fixed by the anti-automorphism of FG which inverts all elements of G. It is known that U is nilpotent if G is nilpotent and the commutator subgroup G′ has p-power order, and then the nilpotency class of U is at most the order of G′; this bound is attained if and only if G′ is cyclic and not a Sylow subgroup of G. Adalbert Bovdi and János Kurdics proved the 'if' part of this last statement by exhibiting a nontrivial commutator of the relevant weight. For the special case when G is a nonabelian torsion group (so G′ cannot possibly be a Sylow subgroup), the present paper identifies such a commutator in U+, showing (Theorem 1) that the same bound is attained even by the nilpotency class of this subgroup. We do not know what happens when G′ is not a Sylow subgroup but G is not torsion. It can happen that U+ is nilpotent even though U is not. The torsion groups G which allow this are known (from the work of Gregory T. Lee) to be precisely the direct products of a finite p-group P, a quaternion group Q of order 8, and an elementary abelian 2-group. Theorem 2: in this case, the nilpotency class of U+ is strictly smaller than the nilpotency index of the augmentation ideal of the group algebra FP, and this bound is attained whenever P is a powerful p-group. The nonabelian group P of order 27 and exponent 3 is not powerful, yet the G = P × Q formed with this P also leads to a U+ attaining the general bound, so here a necessary and sufficient condition remains elusive.

Original languageEnglish
Pages (from-to)171-180
Number of pages10
JournalPublicationes Mathematicae Debrecen
Issue number1-2
Publication statusPublished - 2011
Externally publishedYes


  • Group ring
  • Involution
  • Nilpotency class
  • Symmetric units

ASJC Scopus subject areas

  • General Mathematics


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