Two theorems in the commutator calculus
HTML articles powered by AMS MathViewer
- by Hermann V. Waldinger PDF
- Trans. Amer. Math. Soc. 167 (1972), 389-397 Request permission
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 $\mathcal {G}’$.References
- S. Bachmuth, Automorphisms of free metabelian groups, Trans. Amer. Math. Soc. 118 (1965), 93–104. MR 180597, DOI 10.1090/S0002-9947-1965-0180597-3
- K. W. Gruenberg, Residual properties of infinite soluble groups, Proc. London Math. Soc. (3) 7 (1957), 29–62. MR 87652, DOI 10.1112/plms/s3-7.1.29
- Marshall Hall Jr., The theory of groups, The Macmillan Company, New York, N.Y., 1959. MR 0103215 Phillip Hall, A contribution to the theory of groups of prime power order, Proc. London Math. Soc. (2) 36 (1934), 29-95.
- Wilhelm Magnus, On a theorem of Marshall Hall, Ann. of Math. (2) 40 (1939), 764–768. MR 262, DOI 10.2307/1968892
- Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
- Ruth Rebekka Struik, On nilpotent products of cyclic groups. II, Canadian J. Math. 13 (1961), 557–568. MR 144946, DOI 10.4153/CJM-1961-045-2
- Hermann V. Waldinger, The lower central series of groups of a special class, J. Algebra 14 (1970), 229–244. MR 260882, DOI 10.1016/0021-8693(70)90124-9
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 167 (1972), 389-397
- MSC: Primary 20F35
- DOI: https://doi.org/10.1090/S0002-9947-1972-0294467-7
- MathSciNet review: 0294467