![]() ![]() A group whose transfinite derived series reaches the trivial group is called a hypoabelian group, and every solvable group is a hypoabelian group.Into the Breach, our game of the year for 2018, is a masterpiece of simplicity. For an infinite group, the finite derived series may not stabilize, but the transfinite derived series always stabilizes. Clearly all solvable groups are virtually solvable, since one can just choose the group itself, which has index 1.Ī solvable group is one whose derived series reaches the trivial subgroup at a finite stage. ![]() Virtually solvable groups Ī group G is called virtually solvable if it has a solvable subgroup of finite index. If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups:Ĭyclic < abelian < nilpotent < supersolvable < polycyclic < solvable < finitely generated group. The alternating group A 4 is an example of a finite solvable group that is not supersolvable. In fact, all supersolvable groups are finitely generated, and an abelian group is supersolvable if and only if it is finitely generated. Since a normal series has finite length by definition, uncountable groups are not supersolvable. The direct product of a sequence of solvable groups with unbounded derived length is not solvable, so the class of all solvable groups is not a variety.Īs a strengthening of solvability, a group G is called supersolvable (or supersoluble) if it has an invariant normal series whose factors are all cyclic.
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