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Solvable group: decomposable in prime order groups?

  1. Sep 17, 2013 #1

    From MathWorld on solvable group:
    But why is that a special case? The way I understand it: the normal series can always be made such that all composition factors are simple, but then the composition factors are both simple and Abelian, and hence (isomorphic to) [itex]\mathbb Z_p[/itex], i.e. the composition index is p (= prime)...
  2. jcsd
  3. Sep 17, 2013 #2


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    It is only for a finite group that you are guaranteed to have a composition series. For an infinite group, there may be no normal series where the subgroups are maximal. For instance, ##\mathbb{Z}## cannot have a composition series, since it is not itself simple (every subgroup of ##\mathbb{Z}## is itself isomorphic to ##\mathbb{Z}##).
  4. Sep 18, 2013 #3
    Thank you! I see, so for a finite group the "special case" is always true; that clarifies!
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