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 | References | Similar Articles | Additional Information

Abstract: Let . Let be the *n*th subgroup of the lower central series. Let *p* be a prime. Let be the basic commutators of dimension but . Let for . Then . It is shown in Theorem 1 that the exponents are divisible by *p*, except for the exponent of which .

Let the group 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 be its commutator subgroup. It is proven in Theorem 2 that the `` -simple basic commutators'' of dimension defined below are free generators of .

**[1]**S. Bachmuth,*Automorphisms of free metabelian groups*, Trans. Amer. Math. Soc.**118**(1965), 93-104. MR**0180597**, 10.1090/S0002-9947-1965-0180597-3**[2]**K. W. Gruenberg,*Residual properties of infinite soluble groups*, Proc. London Math. Soc. (3)**7**(1957), 29–62. MR**0087652****[3]**Marshall Hall Jr.,*The theory of groups*, The Macmillan Co., New York, N.Y., 1959. MR**0103215****[4]**Phillip Hall,*A contribution to the theory of groups of prime power order*, Proc. London Math. Soc. (2)**36**(1934), 29-95.**[5]**Wilhelm Magnus,*On a theorem of Marshall Hall*, Ann. of Math. (2)**40**(1939), 764–768. MR**0000262****[6]**Wilhelm Magnus, Abraham Karrass, and Donald Solitar,*Combinatorial group theory: Presentations of groups in terms of generators and relations*, Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1966. MR**0207802****[7]**Ruth Rebekka Struik,*On nilpotent products of cyclic groups. II*, Canad. J. Math.**13**(1961), 557–568. MR**0144946****[8]**Hermann V. Waldinger,*The lower central series of groups of a special class*, J. Algebra**14**(1970), 229–244. MR**0260882**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0294467-7

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*,
" -simple basic commutator",
" -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