Two theorems in the commutator calculus
Author:
Hermann V. Waldinger
Journal:
Trans. Amer. Math. Soc. 167 (1972), 389397
MSC:
Primary 20F35
MathSciNet review:
0294467
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Abstract: Let . Let be the nth 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 .
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 S. Bachmuth, Automorphisms of free metabelian groups, Trans. Amer. Math. Soc. 118 (1965), 93104. MR 31 #4831. MR 0180597 (31:4831)
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 K. W. Gruenberg, Residual properties of infinite soluble groups, Proc. London Math. Soc. (3) 7 (1957), 2962. MR 19, 386. MR 0087652 (19:386a)
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 Phillip Hall, A contribution to the theory of groups of prime power order, Proc. London Math. Soc. (2) 36 (1934), 2995.
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 Wilhelm Magnus, On a theorem of Marshall Hall, Ann. of Math. 40 (1939), 764768. MR 1, 44. MR 0000262 (1:44b)
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 Wilhelm Magnus, Abraham Karrass and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Pure and Appl. Math., vol. 13, Interscience, New York, 1966. MR 34 #7617. MR 0207802 (34:7617)
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 Ruth Rebekka Struik, On nilpotent products of cyclic groups. II, Canad. J. Math. 13 (1961), 557568. MR 26 #2486. MR 0144946 (26:2486)
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 Hermann V. Waldinger, The lower central series of groups of a special class, J. Algebra 14 (1970), 229244. MR 41 #5502. MR 0260882 (41:5502)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197202944677
PII:
S 00029947(1972)02944677
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
