TOPICS

# Solvable Group

A solvable group is a group having a normal series such that each normal factor is Abelian. The special case of a solvable finite group is a group whose composition indices are all prime numbers. Solvable groups are sometimes called "soluble groups," a turn of phrase that is a source of possible amusement to chemists.

The term "solvable" derives from this type of group's relationship to Galois's theorem, namely that the symmetric group is unsolvable for while it is solvable for , 2, 3, and 4. As a result, the polynomial equations of degree are (in general) not solvable using finite additions, multiplications, divisions, and root extractions.

A major building block for the classification of finite simple groups was the Feit-Thompson theorem, which proved that every group of odd order is solvable. This proof took up an entire journal issue.

Every finite group of order , every Abelian group, and every subgroup of a solvable group is solvable. Betten (1996) has computed a table of solvable groups of order up to 242 (Besche and Eick 1999).

Abelian Group, Composition Series, Feit-Thompson Theorem, Galois Group, Galois's Theorem, Solvable Lie Group, Symmetric Group

## Explore with Wolfram|Alpha

More things to try:

## References

Besche, H.-U. and Eick, B. "The Groups of Order at Most 1000 Except 512 and 768." J. Symb. Comput. 27, 405-413, 1999.Betten, A. "Parallel Construction of Finite Soluble Groups." In Parallel Virtual Machine, Euro PVM '96: Third European PVM Conference, Munich, Germany, October 7-9, 1996 (Ed. A. Bode et al. ). Berlin: Springer-Verlag, pp. 126-133, 1996.Doerk, K. and Hawkes, T. Finite Soluble Groups. Berlin: de Gruyter, 1992.Gruenberg, K. W. and Roseblade, J. E. (Eds.). Group Theory: Essays for Philip Hall. London: Academic Press, 1984.Laue, R. "Zur Konstruktion und Klassifikation endlicher auflösbarer Gruppen." Bayreuther Mathemat. Schriften 9, 1982.Lomont, J. S. Applications of Finite Groups. New York: Dover, p. 26, 1993.Magnus, W. "Neuere Ergebnisse über auflösbare Gruppen." Jahresber. der DMV 47, 69, 1937.Robinson, D. J. S. Finiteness Conditions and Generalized Soluble Groups, 2 vols. Berlin: Springer-Verlag, 1972.Scott, W. R. "Solvable Groups." §2.6 in Group Theory. New York: Dover, pp. 38-39, 1987.Segal, D. Polycyclic Groups. Cambridge, England: Cambridge University Press, 1983.

Solvable Group

## Cite this as:

Weisstein, Eric W. "Solvable Group." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SolvableGroup.html