An improvement on the upper bound of the nilpotency class of semidirect products of 𝑝-groups

Morley, Larry J.; Meldrum, John D. P.

doi:10.1090/S0002-9939-1976-0427470-3

An improvement on the upper bound of the nilpotency class of semidirect products of $p$-groups
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by Larry J. Morley and John D. P. Meldrum PDF

Proc. Amer. Math. Soc. 60 (1976), 53-56 Request permission

Abstract:

The semidirect product of a group A by a group B is necessarily nilpotent only in the case A and B are p-groups for the same prime p, A is nilpotent of bounded exponent, and B is finite. In an earlier paper Morley has established an upper bound on the class of a nilpotent semidirect product of an abelian p-group of bounded exponent by an arbitrary finite p-group. In this paper this result is improved by considering a direct product decomposition for B and also by extending the results to give a new upper bound on the class in the most general case. The standard wreath product of A by B is a nilpotent semidirect product of relatively large class in the case A and B satisfy the conditions above, and this new bound improves the known results on the class of these wreath products.

References

Gilbert Baumslag, Wreath products and $p$-groups, Proc. Cambridge Philos. Soc. 55 (1959), 224–231. MR 105437

Nilpotent groups

Hans Liebeck, Concerning nilpotent wreath products, Proc. Cambridge Philos. Soc. 58 (1962), 443–451. MR 139656
J. D. P. Meldrum, On nilpotent wreath products, Proc. Cambridge Philos. Soc. 68 (1970), 1–15. MR 260880, DOI 10.1017/s0305004100000980
Larry Morley, Bounds on the nilpotency class of certain semidirect products, Trans. Amer. Math. Soc. 159 (1971), 381–390. MR 284512, DOI 10.1090/S0002-9947-1971-0284512-6