Page 117 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.

mož Moravec: Some Topics in the Theory of Finite Groups 105

Derived series, upper and lower central series

Define G = [G ,G ] and inductively G (0) = G and G (n+1) = (G (n)) . The derived series of G is
the series

G (0) G (1) G (2) · · ·

of fully invariant (and therefore normal) subgroups of G . The derived series of a group is
in close connection with its solvability:

Proposition 3.5.6 If 1 = G0 G1 · · · Gn = G is an abelian series of a solvable group G ,
then G (i ) Gn−i and, in particular, G (n) = 1. The derived length of G is equal to the length
of the derived series.

PROOF. We prove this by induction, the case i = 0 being trivial. If the assertion is true for
i , then G (i +1) = (G (i )) (Gn−i ) Gn−i −1, as required.

GAP can compute the derived series as follows:
gap> G := OneSmallGroup(Size, 120, IsAbelian, false, IsSolvable, true);;
gap> StructureDescription(G);
"C5 x (C3 : C8)"
gap> DerivedSeries(G);
[ C5 x (C3 : C8), Group([ f5 ]), Group([ ]) ]
gap> DerivedLength(G);
2

There are two canonical central series of a given group. Define γ1(G ) = G and induc-
tively γn+1(G ) = [γn (G ),G ]. The result is the lower central series

G = γ1(G ) γ2(G ) · · ·

of fully invariant (and therefore normal) subgroups. The factor group γn (G )/γn+1(G ) lies
in the center of G /γn+1(G ).

Define Z0(G ) = 1 and inductively Zn+1(G )/Zn (G ) = Z (G /Zn (G )). We obtain the upper
central series

1 = Z0(G ) Z1(G ) Z2(G ) · · ·

of characteristic (and therefore normal) subgroups of G . If G is finite, it terminates in a
subgroup called the hypercenter of G .

Proposition 3.5.7 If 1 = G0 G1 · · · Gn = G is a central series of a nilpotent group G ,
then

1. γi (G ) Gn−i +1, so that γn+1(G ) = 1;
2. Gi Zi (G )so that Zn (G ) = G ;