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On Central Series

I. D. Macdonald
1962 Proceedings of the Edinburgh Mathematical Society  
Let 1 = Z and be, respectively, the upper and lower central series of a group G. Our purpose in this note is to extend known results and find some information as to which of the factors ZJZ fc _i and F k /r k+1 may be infinite. Though our conclusions about the lower central series will be quite general we assume in the other case that the group is f.n., i.e. an extension of a finite group by a nilpotent group. The essential facts about f.n. groups are to be found in P. Hall's paper (4). We also
more » ... paper (4). We also refer to (4) for general notation; we reserve the letter k for positive integers. We note two well-known theorems, the first of which is proved in both (1) and (2):
doi:10.1017/s0013091500014772 fatcat:ct6juagaxvdrdkthqd7i2bilx4