A note on the upper Lie nilpotency index of a group algebra

Abstract

In this paper, we classify the modular group algebra KG of a group G over a field K of characteristic p > 0 having upper Lie nilpotency index t L (KG) = |G ′ | − k(p − 1) + 1 for k = 14 and 15. Group algebras of upper Lie nilpotency index |G ′ | − k(p − 1) + 1 for k ≤ 13, have already been characterized completely.

On the Lie nilpotency index of modular group algebras

“…, then from the Table 5 of [14], no such group exists. If |ζ(G ′ )| = 27, then possible G ′ are S(243, 2), S(243, 32) to S(243, 36) and S(243, 62) to S(243, 64) (see Table 5 of [14]). Now [12] no such group exists.…”

“…Furthermore, group algebras with minimal Lie nilpotency index p + 1 have been classified by Sharma and Bist [20]. A complete description of the Lie nilpotent group algebras with next possible nilpotency indices 2p, 3p−1, 4p−2, 5p−3, 6p−4, 7p−5, 8p−6 and 9p−7 is given in [13][14][15][16][17].…”

“…is any one of the groups S(32, 46) to S(32, 48), then either [14] the possibilities for G ′ are S(81, 3), S(81, 4), S(81, 12) and S(81, 13), but then (16,4) and S (16,11) to S (16,13). But for these groups…”

“…Then possible G ′ are S(729, 422) to S(729, 424) and S(729, 502) (see Table 6 of [14]). But then |D (3),K (G)| = 3 3 .…”

“…But then |D (3),K (G)| = 3 3 . If |ζ(G ′ )| = 3 3 , then no group exists (see Table 6 of [14]). Let |ζ(G ′ )| = 3 4 .…”