Abstract
We say that a group G has many elliptic pairs of subgroups if every infinite set {H 1, H 2, …} of subgroups of G contains at least one pair H i, Hj, i ≠ j such that <H i, Hj>=(H iHj)n for some integer n depending on H i, Hj. By (H iHj)n we mean the set of all products of 2n elements each from H i or H j. Let G be a finitely generated solvable group. It is shown that G has many elliptic pairs of subgroups if and only if G is finite-by-nilpotent. It is also shown that if G is finitely generated, torsion-free and residually finite p-group, for some prime p, then G has many elliptic pairs of subgroups if and only if G is nilpotent.
Keywords
- Normal Subgroup
- Nilpotent Group
- Solvable Group
- Wreath Product
- Finite Index
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© 1989 Springer-Verlag
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Rhemtulla, A.H. (1989). Groups with many elliptic subgroups. In: Kim, A.C., Neumann, B.H. (eds) Groups — Korea 1988. Lecture Notes in Mathematics, vol 1398. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086252
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DOI: https://doi.org/10.1007/BFb0086252
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