[Definitions: The socle of a group $G$ is the subgroup generated by the minimal normal subgroups of $G$. The term "rank" is meant in the sense of the Mal’cev special or Prüfer rank: by definition a group $H$ has finite rank $\le n$ if every finitely generated subgroup of $X$ can be generated by $\le n$ elements. A group which does not have finite rank is said to have infinite rank. Below $p$ denotes a given prime number.]

The socle of an abelian $p$-group of infinite rank has infinite rank. I wonder if it is true in the locally nilpotent case:

Is the following assertion true: Let $G$ be locally nilpotent $p$-group. If $G$ has infinite rank, then the socle of $G$ has infinite rank.


1 Answer 1


The result is false in general.

A result of A. I. Mal'cev ('On certain classes of infinite soluble groups', translated in Amer. Math. Soc. Translations (2), 2 (1956), 1-21, DOI: http://dx.doi.org/10.1090/trans2/002/01) shows that a locally nilpotent $p$-group with finite Prüfer rank is hypercentral (and satisfies Min). On the other hand there are examples of locally nilpotent $p$-groups which are characteristically simple (see, for example, D.H. McLain, A characteristically-simple group. Proc. Cambridge Philos. Soc. 50 (1954), 641–642, DOI: https://doi.org/10.1017/S0305004100029819 (for a good construction of which you can refer to Robinson's Finiteness Conditions Vol.2), and P. Hall, 'Wreath Powers and Characteristically Simple Groups', Proc. Cambridge Philos. Soc. 58 (1962), 170–184, DOI: https://doi.org/10.1017/S0305004100036379). A group of this kind has infinite rank by the quoted result of Mal'cev, but has trivial socle, since it is clearly a characteristic subgroup.


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