# Principal series

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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.