# Principal series

*of length $m$*

A finite descending chain

$$G=G_0>G_1>\dots>G_m=1$$

of normal subgroups of a group $G$ that cannot be included (without repetition) in any other chain with the same properties, i.e. $G_{i+1}$ is a maximal normal subgroup of $G$ contained in $G_i$ as a proper subgroup, $i=0,\dots,m-1$. A group has at least one principal series if and only if all ascending and descending chains of normal subgroups have finite length. If a group has two principal series, then they are isomorphic, i.e. they have the same length and there exists a bijection between the set of quotients $G_i/G_{i+1}$ of one series and the set of quotients of the other series, corresponding factors being isomorphic.

#### Comments

The terminology "principal series" is almost never used in the West. Instead one uses chief series. The isomorphism statement above is the Jordan–Hölder theorem for chief series. The quotients $G_i/G_{i+1}$ defined by a chief series are called chief factors. Any chief series can be refined to a composition series (cf. Composition sequence).

#### References

[a1] | R. Carmichael, "Groups of finite order" , Dover, reprint (1956) pp. 97 |

[a2] | M. Hall jr., "The theory of groups" , Macmillan (1959) pp. 124 |

[a3] | B. Huppert, "Endliche Gruppen" , 1 , Springer (1967) pp. 64 |

**How to Cite This Entry:**

Principal series.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Principal_series&oldid=43601