A note on normal complements in mod 𝑝 envelopes

Abstract

Let G be a finite p-group and let Z p [ G ] {Z_p}[G] denote the group ring of G over the field of p elements. The mod p \bmod \;p envelope of G, denoted by G ∗ {G^ \ast } , is the set of elements of Z p [ G ] {Z_p}[G] with coefficient-sum equal to 1. Many examples of p-groups that have a normal complement in G ∗ {G^ \ast } have been found, including ten of the fourteen different groups of order 16. This note proves that one of the remaining groups of order 16 has a normal complement. The remaining groups of order 16 are the dihedral, semidihedral, and generalized quaternion groups of order 2 n , n = 4 {2^n},n = 4 . We will prove that these groups do not have a normal complement for any n ⩾ 4 n \geqslant 4 .