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

$$I\subset H_s \subset \ldots \subset H_2 \subset H_1 \subset G$$

is one Composition Series and

$$I\subset K_t \subset \ldots \subset K_2 \subset K_1 \subset G$$

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

$${K_j\over K_{j+1}}={H_i\over H_{i+1}}.$$

This theorem was proven in 1869-1889.

See also Composition Series, Finite Group, Isomorphic Groups

References

Lomont, J. S. Applications of Finite Groups. New York: Dover, p. 26, 1993.