Abstract
The question implied in the title is a problem of A. Zaks, namely, which finite groups (other than abelian and simple groups) have all their proper factors abelian? This paper answers the question in the case of groups with non-trivial centre, or, equivalently, in the case ofp-groups, and gives a structure theorem for such groups.
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Reference
P. Hall,A contribution to the theory of groups of prime-power order Proc. London Math. Soc. (1933), 29–95.
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Klein, T. Groups whose proper factors are all abelian. Israel J. Math. 9, 362–366 (1971). https://doi.org/10.1007/BF02771686
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DOI: https://doi.org/10.1007/BF02771686