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Jordan-Hölder Theorem

The Jordan-Hölder theorem states that the composition quotient groups belonging to two composition series of a finite group G are, apart from their sequence, isomorphic in pairs. Put H_0=K_0=G and H_(s+1)=K_(t+1)=I. In other words, if

I subset H_s subset ... subset H_2 subset H_1 subset G
(1)

is one composition series and

I subset K_t subset ... subset K_2 subset K_1 subset G
(2)

is another, then t=s, and corresponding to any composition quotient group K_j/K_(j+1), there is a composition quotient group H_i/H_(i+1) such that

(K_j)/(K_(j+1))=(H_i)/(H_(i+1)).
(3)

This theorem was proven in 1869-1889.

The Jordan-Hölder theorem also applies to modules with a composition series. Any two such series have the same length, and their successive quotient modules are isomorphic up to order (Atiyah and Macdonald 1969, p. 77).

See also

Butterfly Lemma, Composition Series, Finite Group, Isomorphic Groups, Module, Module Length, Quotient Module

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References

Atiyah, M. F. and Macdonald, I. G. Introduction to Commutative Algebra. Menlo Park, CA: Addison-Wesley, pp. 76-78, 1969.Lomont, J. S. Applications of Finite Groups. New York: Dover, p. 26, 1993.Scott, W. R. §2.5.8 in Group Theory. New York: Dover, p. 37, 1987.

Referenced on Wolfram|Alpha

Jordan-Hölder Theorem

Cite this as:

Weisstein, Eric W. "Jordan-Hölder Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Jordan-HoelderTheorem.html

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