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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Two theorems in the commutator calculus

Author: Hermann V. Waldinger
Journal: Trans. Amer. Math. Soc. 167 (1972), 389-397
MSC: Primary 20F35
MathSciNet review: 0294467
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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}'$.

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Keywords: Group, subgroup, free product of groups, direct product of groups cyclic group, abelian group, free group, free abelian group, finitely generated group, associative algebra, generator, free generator, relator, order of group, prime p, commutator, basic commutator, commutator subgroup, 2nd commutator subgroup, lower central series, subgroup of the lower central series, dimension, indeterminate, commuting indeterminates, formal infinite sum, homogeneous polynomial, degree of a polynomial, linear independence, integral coefficient, coefficient or exponent divisible by p, " $ \mathcal{F}$-simple basic commutator", " $ \mathcal{G}$-simple basic commutator", abelianized group, faithful representation, collection process, nontrivial relator, Jacobi relator, homomorphism, image under a homomorphism
Article copyright: © Copyright 1972 American Mathematical Society