Solvable Group

A solvable group is a group whose composition indices are all Prime Numbers. Equivalently, a solvable group is a Group having a ``normal series'' such that each ``normal factor'' is Abelian. The term solvable derives from this type of group's relationship to Galois's Theorem, namely that the Symmetric Group $S_n$ is insoluble for $n\geq 5$ while it is solvable for $n=1$, 2, 3, and 4. As a result, the Polynomial equations of degree $\geq 5$ are not solvable using finite additions, multiplications, divisions, and root extractions.

Every Finite Group of order $<60$, every Abelian Group, and every Subgroup of a solvable group is solvable.

See also Abelian Group, Composition Series, Galois's Theorem, Symmetric Group

References

Lomont, J. S. Applications of Finite Groups. New York: Dover, p. 26, 1993.