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 S_n is unsolvable for n>=5 while it is solvable for n=1, 2, 3, and 4. As a result, the polynomial equations of degree >=5 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 <60, 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).

See also

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

Explore with Wolfram|Alpha


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.

Referenced on Wolfram|Alpha

Solvable Group

Cite this as:

Weisstein, Eric W. "Solvable Group." From MathWorld--A Wolfram Web Resource.

Subject classifications