Two theorems in the commutator calculus

Abstract:

Let $F = \langle a,b\rangle$. Let ${F_n}$ be the nth subgroup of the lower central series. Let p be a prime. Let ${c_3} < {c_4} < \cdots < {c_z}$ be the basic commutators of dimension $> 1$ but $< p + 2$. Let ${P_1} = (a,b),{P_m} = ({P_{m - 1}},b)$ for $m > 1$. Then $(a,{b^p}) \equiv \prod \nolimits _{i = 3}^z {c_i^{{\eta _i}}\bmod {F_{p + 2}}}$. It is shown in Theorem 1 that the exponents ${\eta _i}$ are divisible by p, except for the exponent of ${P_p}$ which $= 1$. Let the group $\mathcal {G}$ 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 $\mathcal {G}’$ be its commutator subgroup. It is proven in Theorem 2 that the “$\mathcal {G}$-simple basic commutators” of dimension $> 1$ defined below are free generators of $\mathcal {G}’$.