Supersolvable group

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Short description: Group with series of normal subgroups where all factors are cyclic

In mathematics, a group is supersolvable (or supersoluble) if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvability.


Let G be a group. G is supersolvable if there exists a normal series

[math]\displaystyle{ \{1\} = H_0 \triangleleft H_1 \triangleleft \cdots \triangleleft H_{s-1} \triangleleft H_s = G }[/math]

such that each quotient group [math]\displaystyle{ H_{i+1}/H_i \; }[/math] is cyclic and each [math]\displaystyle{ H_i }[/math] is normal in [math]\displaystyle{ G }[/math].

By contrast, for a solvable group the definition requires each quotient to be abelian. In another direction, a polycyclic group must have a subnormal series with each quotient cyclic, but there is no requirement that each [math]\displaystyle{ H_i }[/math] be normal in [math]\displaystyle{ G }[/math]. As every finite solvable group is polycyclic, this can be seen as one of the key differences between the definitions. For a concrete example, the alternating group on four points, [math]\displaystyle{ A_4 }[/math], is solvable but not supersolvable.

Basic Properties

Some facts about supersolvable groups:

  • Supersolvable groups are always polycyclic, and hence solvable.
  • Every finitely generated nilpotent group is supersolvable.
  • Every metacyclic group is supersolvable.
  • The commutator subgroup of a supersolvable group is nilpotent.
  • Subgroups and quotient groups of supersolvable groups are supersolvable.
  • A finite supersolvable group has an invariant normal series with each factor cyclic of prime order.
  • In fact, the primes can be chosen in a nice order: For every prime p, and for π the set of primes greater than p, a finite supersolvable group has a unique Hall π-subgroup. Such groups are sometimes called ordered Sylow tower groups.
  • Every group of square-free order, and every group with cyclic Sylow subgroups (a Z-group), is supersolvable.
  • Every irreducible complex representation of a finite supersolvable group is monomial, that is, induced from a linear character of a subgroup. In other words, every finite supersolvable group is a monomial group.
  • Every maximal subgroup in a supersolvable group has prime index.
  • A finite group is supersolvable if and only if every maximal subgroup has prime index.
  • A finite group is supersolvable if and only if every maximal chain of subgroups has the same length. This is important to those interested in the lattice of subgroups of a group, and is sometimes called the Jordan–Dedekind chain condition.
  • Moreover, a finite group is supersolvable if and only if its lattice of subgroups is a supersolvable lattice, a significant strengthening of the Jordan-Dedekind chain condition.
  • By Baum's theorem, every supersolvable finite group has a DFT algorithm running in time O(n log n).[clarification needed]