TOPICS
Search

Jordan-Hölder Theorem


The composition quotient groups belonging to two composition series of a finite group G are, apart from their sequence, isomorphic in pairs. 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.


See also

Butterfly Lemma, Composition Series, Finite Group, Isomorphic Groups

Explore with Wolfram|Alpha

References

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 Web Resource. https://mathworld.wolfram.com/Jordan-HoelderTheorem.html

Subject classifications