Integral group rings of solvable groups with trivial central units

Abstract

The integral group ring $\mathbb{Z} G$ of a group $G$ has only trivial central units, if the only central units of $\mathbb{Z} G$ are $\pm z$ for $z$ in the center of $G$. We show that the order of a finite solvable group $G$ with this property, can only have $2$, $3$, $5$ and $7$ as prime divisors, by linking this to inverse semi-rational groups and extending one result on this class of groups. We also classify the Frobenius groups whose integral group rings have only trivial central units.Comment: [v4]: 13 pages. Final version. To appear in "Forum Mathematicum" (already available online

Finite simple groups with short Galois orbits on conjugacy classes

“…Feit and Seitz [17] proved that the only rational non-abelian finite simple groups are Sp(6, 2) and SO + (8,2). In [1] a list of groups for which f (G) ≤ 2 was given.…”

“…, PSL(2, 11), PSL (3,4), PSp(6, 3), Sp (8,2), SU(3, 3), PSU (3,5), SU(4, 2) ∼ = PSp(4, 3), PSU(4, 3), SU(5, 2), PSU(6, 2), PΩ(7, 3), PΩ + (8, 3), F 4 (2), G 2 (3), G 2 (4), 2 E 6 (2)} or G is one of twenty sporadic simple groups different from Ly and different from the five sporadic groups listed in (iii). (2,13), PSL (2,17), PSL (2,19), Sp(10, 2), PSp(4, 5), 3…”

“…(iv) f (G) = 4 if and only if G ∈ { C 5 , PSL(2, 29), PSL(2, 31), SL(3, 3), PSL(4, 3), Sp (12,2), Sp(4, 4), SU (3,4), Ω − (8,2), Ω − (10, 2), Ω + (12, 2), 2 F 4 (2) ′ }.…”

“…LetS be any of the groups listed in the statement of Proposition 5.3. IfS = PSL(n, q) is (non-abelian) simple and different from an alternating group, then 4n < ϕ (q n − 1)/(d(q − 1)) where d = (n, q − 1) unless (n, q) is in { (2, 7), (2,8), (2,11), (2,13), (2,17), (2,19), (2,23), (2,27), (2, 29), (2, 31), LetS = PSp(n, q) be simple with n ≥ 4 even. This group contains a cyclic subgroup of order 1 d (q n/2 + 1) where d = (2, q − 1).…”

“…The first results on cut-groups were obtained by A. Bovdi and Patay. Recent results and references on this topic can be found in [8,10] and [11].…”