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The fourth dimension subgroups and polynomial maps, II

Published online by Cambridge University Press:  22 January 2016

Ken-Ichi Tahara
Ken-Ichi Tahara*
Affiliation:
Aichi University of Education
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In our previous paper [3] we proved the following ([3, Theorem 16]) :

THEOREM A. Let G be a 2-group of class 3. Let G2 and G/G2 be direct products of cyclic groups 〈yq〉 of order αq (1 ≦ q ≦ m), and of cyclic groups 〈hi〉 of order βi (1 ≦ i ≦ n) with β1 ≧ β2 ≧ · · · βn, respectively. Let xi be representatives of hi (1 ≦ i ≦ n), and put Then a homomorphism ψ:G3→T can be extended to a polynomial map from G to T of degree ≦ 4 if and only if there exists an integral solution in the following linear equations of Xiq (1 ≦ i ≦ n, 1 ≦ q ≦ m) with coefficients in T:

(I)

where δij is the Kronecker symbol for βi: i.e. δij = 1 or 0 according to βi = βj or βi > βj, respectively.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 69 , February 1978 , pp. 1 - 7
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1978

References

[1] Passi, I. B. S., Dimension subgroups, J. Algebra 9 (1968), 152–182.CrossRefGoogle Scholar
[2] Rips, E., On the fourth integer dimension subgroup, Israel J. Math. 12 (1972), 342–346.Google Scholar
[3] Tahara, K., The fourth dimension subgroups and polynomial maps, J. Algebra 45 (1977), 102–131.Google Scholar
[4] Tahara, K., On the structure of Qs(G) and the fourth dimension subgroups, Japan. J. Math. 3 (New Ser) (1977).CrossRefGoogle Scholar
[5] Tahara, K., Some remarks on the structure of Qn(G) for finitely generated groups G, to appear.Google Scholar