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Let G be a solvable group. We prove that if G has a composition series, then G has to be finite. (Note that this is sort of a converse to “A finite group has a composition series.”)
Let be a composition series of , where each factor is simple.
Since and are solvable (every subgroup of a solvable group is solvable), the quotient is also solvable.
We can prove that is abelian. Since , by the fact that the factor is simple, we have or .
If , then this contradicts the fact that is solvable. Thus and is abelian.
Key step: is simple and abelian, for some prime .
Since , so we have that . By induction, .
. Thus G is finite.