Algebra I: Section 6. The structure of groups. 6.1 Direct products of ...

6.4.10 Lemma. A group G is solvable if there exist subgroups
such that
G = G0 ⊇ G1 ⊇ . . . ⊇ Gr ⊇ Gr+1 = (e)
(i) Gj+1 is a normal subgroup in Gj for each j,
(ii) each quotient Gj/Gj+1 is solvable.
In particular G is solvable if the quotients are all abelian. Conversely, if G is solvable such
sequences exist and the subgroups can be chosen so each quotient is abelian.
Proof: The previous remarks show that we can work backward up the chain G ⊇ D1(G) ⊇
D2(G) ⊇ . . ., successively verifying solvability of Gr, Gr−1, . . .. Starting from the top, Exercise
6.4.3 shows that G/D1(G) = G/[G, G] is abelian, but since D2(G) = D1(D1(G)) =
[D1(G), D1(G)] we see that D1(G)/D2(G) is abelian, etc. �
Nilpotent groups, unlike solvable groups, always have nontrivial centers. In fact, if D r (G) �= (e)
and D r+1 (G) = [G, D r (G)] = (e), we see that D r (G) is a nontrivial central subgroup in G.
Central subgroups play a prominent role in the nilpotent analog of 6.4.10.
6.4.11 Lemma. A group G is nilpotent if there there is a subgroup N such that (i) N is central
in G, and (ii) G/N is nilpotent.
Proof: All subgroups of the center Z(Gj) are normal in Gj, so G/N is a bona-fide group.
Now observe that D r+1 (G/N) = (e) for some r and that D r+1 (G/N) = π(D r+1 (G)) under
the quotient map π. Thus D r+1 (G) ⊆ N = kerπ, and since N is central in G we get
D r+2 (G) = [G, D r+1 (G)] ⊆ [G, N] = (e). Thus G is nilpotent. �
For other conditions leading to nilpotence of G see 6.4.13 below.
One useful consequence of 6.4.11 is the fact that nonabelian finite groups always include
sizeable nilpotent subgroups, namely their Sylow p-subgroups.
6.4.12 Corollary. If |G| = p k for some prime then G is nilpotent.
Proof: If k = 1 then G ∼ = Zp, so assume k > 1. Then the center Z(G) is nontrivial and
|G/Z(G)| = p r for some r < k. By induction G/Z(G) is nilpotent, and since Z(G) is central G
must be nilpotent too. �
Nilpotent groups are the next best thing to being abelian; solvable groups also have some
commutative aspects, but the connection is more tenuous. The pervasive role of “centers” in
nilpotent groups is revealed by noting that there is a third canonical series of subgroups in any
group G, the ascending central series Z(G) ⊆ Z (2) ⊆ Z (3) ⊆ . . . defined as follows.
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Z (1) = Z(G) (the center of G)
Z (2) = {x ∈ G : xy ≡ yx (mod Z(G)), all y ∈ G}
Z (3) = {x ∈ G : xy ≡ yx (mod Z (2) ), all y ∈ G}
.
.
Z (k+1) = {x ∈ G : xy ≡ yx (mod Z (k) )}
.
The group Z (2) is often referred to as the second center of G; it is just the pullback to G of
the center in the quotient group G/Z(G). (The center of the quotient need not be trivial.)
Obviously, Z (k) ⊆ Z (k+1) at every step, and if equality holds at the k th step it holds at every
later step. We note without proof the following characterization of nilpotent groups in terms
of the ascending central series.
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