A note on normal complements in mod $p$ envelopes

Abstract:

Let G be a finite p-group and let ${Z_p}[G]$ denote the group ring of G over the field of p elements. The $\bmod \;p$ envelope of G, denoted by ${G^ \ast }$, is the set of elements of ${Z_p}[G]$ with coefficient-sum equal to 1. Many examples of p-groups that have a normal complement in ${G^ \ast }$ 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 ${2^n},n = 4$. We will prove that these groups do not have a normal complement for any $n \geqslant 4$.