Two theorems in the commutator calculus

Abstract

Let F = ⟨ a , b ⟩ F = \langle a,b\rangle . Let F n {F_n} be the nth subgroup of the lower central series. Let p be a prime. Let c 3 > c 4 > ⋯ > c z {c_3} > {c_4} > \cdots > {c_z} be the basic commutators of dimension > 1 > 1 but > p + 2 > p + 2 . Let P 1 = ( a , b ) , P m = ( P m − 1 , b ) {P_1} = (a,b),{P_m} = ({P_{m - 1}},b) for m > 1 m > 1 . Then ( a , b p ) ≡ ∏ i = 3 z c i η i mod F p + 2 (a,{b^p}) \equiv \prod \nolimits _{i = 3}^z {c_i^{{\eta _i}}\bmod {F_{p + 2}}} . It is shown in Theorem 1 that the exponents η i {\eta _i} are divisible by p, except for the exponent of P p {P_p} which = 1 = 1 . Let the group G \mathcal {G} be a free product of finitely many groups each of which is a direct product of finitely many groups of order p, a prime. Let G ′ \mathcal {G}’ be its commutator subgroup. It is proven in Theorem 2 that the “ G \mathcal {G} -simple basic commutators” of dimension > 1 > 1 defined below are free generators of G ′ \mathcal {G}’ .