Definitions: All groups referred to are finite solvable. Call such a group good if it can be constructed from the trivial group using central extensions and split extensions, call a group bad if it can not be so constructed. Call a group ugly if it is centerless and non-split. So, an ugly group is a finite solvable group with trivial center, and which does not have a nontrivial decomposition as a semi-direct product. Can an ugly group have derived length 3? Can an ugly group have 4-th power free order?
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$\begingroup$ 1) A bad group must involve an ugly group, as a quotient of some subgroup. 2) By a result of Curran, a centerless group G of derived length 2 splits over G', so groups of derived length 2 can not be ugly, and must in fact be good. On ther other hand, D Holt used GAP to find ugly groups of derived length 4. Thus the question about derived length 3. $\endgroup$– moshe noimanCommented Jun 19, 2015 at 8:37
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$\begingroup$ 3) Wreath products of cyclic groups are good, by definition, so by Kaloujnine-Krasner all (finite solvable) groups are subgroups of good groups. So goodness is not quotient-closed. 4) nilpotent, more generally supersolvable, more generally Sylow tower groups, are all good. Can the class be made larger without destroying subgroup-closed? $\endgroup$– moshe noimanCommented Jun 19, 2015 at 8:38
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3$\begingroup$ Do you have a reference for the construction of ugly groups by D Holt? (Was it Hardy who said that there was no place in this world for ugly mathematics?) – $\endgroup$– Derek HoltCommented Jun 19, 2015 at 12:52
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1$\begingroup$ You have a definition of ugly groups, and two questions about ugly groups. Before this there is a definition of bad/good groups, with no question about them. Why did you introduce the definition of good/bad groups with no question about them? $\endgroup$– YCorCommented Jun 19, 2015 at 18:13
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$\begingroup$ The definitions of good and bad groups are motivation for the definition of an ugly group. Ugly groups are obstacles to goodness. $\endgroup$– moshe noimanCommented Jun 20, 2015 at 19:32
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