On central unit groups of integral group rings of alternating groups

Finite simple groups with short Galois orbits on conjugacy classes

“…This group contains a cyclic subgroup of order divisible by 1 d (q (n−2)/2 + 1). We have ϕ 1 d (q (n−2)/2 + 1) > 8(n − 2) by the proof of Lemma 5.4, where d = (2, q + 1) unless (n, q) ∈ {(8, 2), (8,3), (8,4), (8,5), (10,2), (10,3), (12,2), (12,3), (14,2), (16,2)}.…”

“…, 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) ′ }.…”

“…Bovdi) was the first to study the question of when a finite simple group is a cut-group (he answered this for alternating groups). Afterwards, several people continued to investigate this question and for a long time the most extensive results on this topic were obtained by Aleev and his students (see [2,3,4,5,5,25]). The (finite) list of finite simple cut-groups was deduced in [9,Theorem 5.1] from a longer list of groups obtained in [1].…”

“…47),(2, 59),(3,2),(3,3),(3,4), (4, 3) }. All projective special linear groups with such exceptional parameters appear in the list of Theorem 1.3 (and thus all Galois orbits on the set of conjugacy classes have size at most 4 by Section 3), apart fromS = PSL(2, 23), PSL(2,27), PSL(2, 47), PSL(2, 59) in which cases f (S) > 4 by Proposition 3.1.…”