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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.