Composition Series

Every finite group G of order greater than one possesses a finite series of subgroups, called a composition series, such that


where H_(i+1) is a maximal subgroup of H_i and H<|G means that H is a normal subgroup of G. A composition series is therefore a normal series without repetition whose factors are all simple (Scott 1987, p. 36).

The quotient groups G/H_1, H_1/H_2, ..., H_(s-1)/H_s, H_s are called composition quotient groups.

See also

Butterfly Lemma, Finite Group, Jordan-Hölder Theorem, Normal Series, Normal Subgroup, Quotient Group, Subgroup

Explore with Wolfram|Alpha


Lomont, J. S. Applications of Finite Groups. New York: Dover, p. 26, 1993.Scott, W. R. "Composition Series." §2.5 in Group Theory. New York: Dover, pp. 36-38, 1987.

Referenced on Wolfram|Alpha

Composition Series

Cite this as:

Weisstein, Eric W. "Composition Series." From MathWorld--A Wolfram Web Resource.

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