The Derived Series of a p-Group

Citation

Hobby, Charles Ray (1960) The Derived Series of a p-Group. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/QY7G-Q706. https://resolver.caltech.edu/CaltechETD:etd-06152006-083527

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Olga Taussky (see W. Magnus, Math. Ann. vol. 111 (1935)) posed the problem of determining whether there is an infinite chain of p-groups G1, G2,..., such that G1 is abelian, [...], and [...] where [...] is the nth derived group of [...]. N. Ito (Nagoya Math. J., vol. 1, (1950)) constructed such a chain for p > 2 and G1 of type (p,p,p). It is shown (by an explicit construction) that if p > 2 there is a chain of the required kind for G1 any non-cyclic abelian p-group. If p = 2 there is a chain of the required kind if G1 contains a subgroup of type [...], of type [...], of type [...], or of type (2,2,2,2,2). As a consequence, for p > 2 it is impossible to estimate the length of the derived series of a non-abelian p-group G from the type of [...]. This gives a negative answer (for p > 2) to a question posed by O. Taussky (Research Problem 9, Bull. Amer. Math. Soc. vol. 64 (1958) pp. 124).

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