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## Main Question or Discussion Point

We have the 3 equivalent definition for solvable groups:

There exists a chain of subgroups

1 < G1 ....< Gi + < G i+1 < Gr = G

such that Gi is normal in Gi+1 and Gi+1/Gi is abelian.

Another definition is

there exists

1 < H1 ....< Hi + < H i+1 < Hs = H

such that Hi is normal in Hi+1, and Hi+1/Hi is cyclic

, the last definition is similar but the quotient group is isomorphic to Z/p.

so my question is, does s need to be equal r?

thanks

There exists a chain of subgroups

1 < G1 ....< Gi + < G i+1 < Gr = G

such that Gi is normal in Gi+1 and Gi+1/Gi is abelian.

Another definition is

there exists

1 < H1 ....< Hi + < H i+1 < Hs = H

such that Hi is normal in Hi+1, and Hi+1/Hi is cyclic

, the last definition is similar but the quotient group is isomorphic to Z/p.

so my question is, does s need to be equal r?

thanks