A note on normal complements in mod envelopes
Abstract: Let G be a finite p-group and let denote the group ring of G over the field of p elements. The envelope of G, denoted by , is the set of elements of with coefficient-sum equal to 1. Many examples of p-groups that have a normal complement in 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 . We will prove that these groups do not have a normal complement for any .
-  D. L. Johnson, The modular group-ring of a finite 𝑝-group, Proc. Amer. Math. Soc. 68 (1978), no. 1, 19–22. MR 0457539, https://doi.org/10.1090/S0002-9939-1978-0457539-0
-  L. E. Moran and R. N. Tench, Normal complements in 𝑚𝑜𝑑𝑝-envelopes, Israel J. Math. 27 (1977), no. 3-4, 331–338. MR 0447403, https://doi.org/10.1007/BF02756491