Geodesic
On the Maximum and Minimum Chain Conditions for the "largeness" Ordering on the Class of Groups
Publications de l'Institut Mathématique, _N_S_37 (1985) no. 51, p. 57

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In previous papers the autor has defined a quasi-order $\preceq$ on the class of groups (the``largeness'' ordering). One can then define the {\it height} of group, and also define what it means for a group to satisfy max-$\preceq$ or min-$\preceq$. A natural question is whether the finiteness conditions max-$\preceq$, min-$\preceq$, ``having finite height'' are extension closed. It is shown here that the answer is ``no'' for all three properties: there is a group which is a split extension of one group of height 1 by another group of height 1, and which does not satisfy max-$\preceq$ or min-$\preceq$.
Classification : 20F22 20E22 
@article{PIM_1985_N_S_37_51_a10,
     author = {S.J. Pride},
     title = {On the {Maximum} and {Minimum} {Chain} {Conditions} for the "largeness" {Ordering} on the {Class} of {Groups}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {57 },
     publisher = {mathdoc},
     volume = {_N_S_37},
     number = {51},
     year = {1985},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1985_N_S_37_51_a10/}
}
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S.J. Pride. On the Maximum and Minimum Chain Conditions for the "largeness" Ordering on the Class of Groups. Publications de l'Institut Mathématique, _N_S_37 (1985) no. 51, p. 57 . http://geodesic.mathdoc.fr/item/PIM_1985_N_S_37_51_a10/