Finite groups are solvable if they have no nontrivial perfect subgroup. But I am sure that for infinite groups, the two notions diverge. Is there standard terminology for groups with no perfect subgroups?
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3$\begingroup$ Every subgroup of a free group is free, in particular non-perfect. But it certainly isn't nilpotent. So, yes, the two notions do indeed diverge. $\endgroup$– HJRWCommented Oct 12, 2010 at 2:21
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1$\begingroup$ When you say "nilpotent" do you mean "solvable"? The group of permutations of three objects is not nilpotent, but it has no perfect subgroups (unless you count the trivial subgroup). $\endgroup$– S. Carnahan ♦Commented Oct 12, 2010 at 2:25
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$\begingroup$ thanks: changed nilpotent to solvable. (and of course I don't count the trivial subgroup). $\endgroup$– Jeff StromCommented Oct 12, 2010 at 2:35
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2$\begingroup$ @Tom: The groups with $G_\lambda=1$ are called hypoabelian. See, for example, Vovsi, S. M. Two notes on local properties of groups. Simon Stevin 55 (1981), no. 1-2, 27–35. Also: planetmath.org/… $\endgroup$– user6976Commented Oct 12, 2010 at 9:59
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2$\begingroup$ @TomGoodwillie the maximal perfect subgroup ("perfect radical"?) exists with no need to define this transfinite derived series. Namely it is the subgroup generated by all perfect subgroups. $\endgroup$– YCorCommented Jul 10, 2022 at 20:50
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1 Answer
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In the infinite case, there is a close notion of "locally indicable group", i.e. a group where every finitely generated subgroup maps onto $\mathbb Z$ (see, for example, this paper). Locally indicable groups are left (right) orderable, hence important. Note that in that notion, not all subgroups are considered but only finitely generated, and "non-perfect" is replaced by a stronger property "maps onto $\mathbb Z$". But in the finite case all subgroups are finitely generated, and "maps onto $\mathbb Z$" is an infinite analog of "maps onto a finite cyclic group" (= non-perfect). So "locally indicable" is possibly the infinite analog of the property you consider.
Update: The groups without perfect subgroups are called hypoabelian. See this text.
