Are there any uncountable soluble groups satisfying Min-n (the minimal condition on normal subgroups) other than the group constructed by B. Hartley in [Uncountable Artinian modules and uncountable soluble groups satisfying Min-n, Proc. London. Math. Soc. (3) 35 (1977) 55-75]?
$\begingroup$
$\endgroup$
6
-
$\begingroup$ Certainly for obvious reasons (e.g. take direct product with a min-n, eg finite, solvable group, most likely yields another group). It's hard to give a better answer without the explicit construction and without knowing what you mean by "other than". $\endgroup$– YCorCommented Jan 20, 2017 at 18:35
-
$\begingroup$ I mean "different from" .... $\endgroup$– Ahmet ArikanCommented Jan 20, 2017 at 18:39
-
$\begingroup$ "different from" is not a mathematical term (or if meant strictly, is certainly inaccurate since you can have non-equal isomorphic groups, e.g. singletons). So you seem to mean "not isomorphic to". Hence by my remark on direct products, the answer is certainly "no" for some trivial reason although as I said, it's hard to be more precise without any hint on the construction (I'd guess it's something akin to a semidirect product $V\rtimes H$ where $V$ is an uncountable abelian $p$-group for some prime $p$, and $H$ being some artinian abelian group???). $\endgroup$– YCorCommented Jan 20, 2017 at 18:46
-
$\begingroup$ I could access Hartley's article... you can't really refer to the group he constructed. It's a family of groups using 3 primes $p,q,r$ (with some condition). They have the form $V\rtimes H$ where $V$ is an artinian $(Z/rZ)H$-module, and $H$ is a semidirect product $C_{p^\infty}\ltimes K$ where $K$ is some subfield of the algebraic closure of $Z/pZ$... There are some papers quoting Hartley, I haven't seen if there are very different examples (other than playing with direct products and little constructions). E.g. I don't know if there's a torsion-free example (I may have missed it). $\endgroup$– YCorCommented Jan 20, 2017 at 21:02
-
$\begingroup$ Thank you. I have not read the paper well, but on p. 167 of [M. R. Dixon, Sylow Theory, Formations and Fitting Classes in Locally Finite Groups] mentions the construction. I wonder if there is a non-isomorphic construction to Hartley's. $\endgroup$– Ahmet ArikanCommented Jan 21, 2017 at 6:06
|
Show 1 more comment