Normal and composition series
A composition series of \(G\) is a chain \(1=G_0\normal G_1\normal\cdots\normal G_n=G\) whose factors \(G_{i+1}/G_i\) are simple groups. By the Jordan–Hölder theorem, any two composition series of a finite group have the same length and the same composition factors, counted with multiplicity and up to isomorphism.
Normal series (Tag D-KM2DV)
Let \(G\) be a group. A normal series of \(G\) is a chain of subgroups \[ G = G^0 \supseteq G^1 \supseteq G^2 \supseteq \cdots \] such that \(G^{i+1}\) is a normal subgroup of \(G^i\) for every \(i\).
Central series (Tag D-5NV3N)
Let \(G\) be a group with identity \(e\). A central series of \(G\) is a finite chain of subgroups \[ G = G^0 \geq G^1 \geq \cdots \geq G^n = \theset{e} \] with each \(G^i \normal G\) and \([G, G^i] \leq G^{i+1}\) for \(0\leq i < n\), where \([G, G^i]\) is the subgroup generated by the commutators \(ghg^{-1}h^{-1}\) with \(g\in G\) and \(h\in G^i\).
For subgroups \(G^{i+1}\leq G^i\) normal in \(G\), the condition \([G, G^i] \leq G^{i+1}\) holds if and only if \(G^i/G^{i+1}\) is contained in the center of \(G/G^{i+1}\).
Composition series (Tag D-FYX4I)
Let \(G\) be a group with identity \(e\). A composition series of \(G\) is a finite normal series \[ G = G^0 \geq G^1 \geq \cdots \geq G^n = \theset{e}, \qquad G^{i+1}\normal G^i, \] such that for each \(0\leq i<n\), \(G^{i+1}\) is a maximal proper normal subgroup of \(G^i\).
Composition series of a group (Tag FD-PLAEO)
Let \(G\) be a group. A composition series of \(G\) is a finite sequence of subgroups \[ 1=N_{0} \leq N_{1} \leq \cdots \leq N_{k-1} \leq N_{k}=G \] such that \(N_i \normal N_{i+1}\) and \(N_{i+1}/N_i\) is simple for \(0\leq i<k\). The quotients \(N_{i+1}/N_i\) are the composition factors of the series.
Jordan--Hölder theorem (Tag T-OSJ5S)
Let \(G\) be a group, and let \[ G=G_0 \trianglerighteq G_1 \trianglerighteq \cdots \trianglerighteq G_r=\theset{e} \quad\text{and}\quad G=H_0 \trianglerighteq H_1 \trianglerighteq \cdots \trianglerighteq H_s=\theset{e} \] be two composition series of \(G\). Then \(r=s\), and there is a permutation \(\pi\) of \(\theset{1,\ldots,r}\) such that \(G_{i-1}/G_i\cong H_{\pi(i)-1}/H_{\pi(i)}\) for every \(i\).
Simple groups (Tag D-T2NZ4)
A group \(G\neq\theset{e}\) is simple if its only normal subgroups are \(\theset{e}\) and \(G\).
Simple group (Tag FD-2UWAQ)
A group \(G\) is simple if \(G\neq\theset{e}\) and its only normal subgroups are the trivial subgroup \(\theset{e}\) and \(G\) [@DF04].
Central and derived series
The lower central series \(\gamma_1(G)=G\), \(\gamma_{i+1}(G)=[\gamma_i(G),G]\) and the upper central series \(Z_0(G)=1\), \(Z_{i+1}(G)/Z_i(G)=Z(G/Z_i(G))\) reach \(1\) and \(G\) respectively if and only if \(G\) is nilpotent, and in that case they have the same length, the nilpotency class of \(G\). The derived series \(G^{(0)}=G\), \(G^{(i+1)}=[G^{(i)},G^{(i)}]\) reaches \(1\) if and only if \(G\) is solvable. Each of these series is determined by \(G\).
Lower central series (Tag D-D7L4X)
Let \(G\) be a group. Set \(G^0 \coloneqq G\) and \(G^{i+1} \coloneqq [G, G^i]\) for \(i\geq 0\), where \([G, G^i]\) is the subgroup generated by the commutators \(ghg^{-1}h^{-1}\) with \(g\in G\) and \(h\in G^i\). The lower central series of \(G\) is the chain \(G^0 \geq G^1 \geq \cdots\).
Lower central series (Tag D-BGNME)
Let \(G\) be a group. Set \(G^0 \coloneqq G\) and \(G^{i+1} \coloneqq [G, G^i]\) for \(i\geq 0\), where \([G, G^i]\) is the subgroup generated by the commutators \(ghg^{-1}h^{-1}\) with \(g\in G\) and \(h\in G^i\). The lower central series of \(G\) is the chain \(G^0 \geq G^1 \geq \cdots\).
The series is descending, while the upper central series is ascending. A group \(G\) with identity \(e\) is nilpotent if and only if \(G^n = \theset{e}\) for some \(n\geq 0\).
Upper central series (Tag D-XEDSI)
Let \(G\) be a group. Define normal subgroups \(Z_i\normal G\) inductively by \(Z_0\coloneqq 1\) and, for \(i\geq 0\), \[ Z_{i+1}\coloneqq \pi_i^{-1}\bigl(Z(G/Z_i)\bigr), \] where \(\pi_i\colon G\to G/Z_i\) is the quotient map and \(Z(G/Z_i)\) is the center of \(G/Z_i\). The upper central series of \(G\) is the chain \(Z_0\leq Z_1\leq Z_2\leq\cdots\).
The induction is well defined: if \(Z_i\normal G\), then \(Z(G/Z_i)\) is normal in \(G/Z_i\), so its preimage \(Z_{i+1}\) is normal in \(G\). \(Z_{i+1}\) is the unique subgroup of \(G\) containing \(Z_i\) with \(Z_{i+1}/Z_i=Z(G/Z_i)\); in particular \(Z_1=Z(G)\).
Derived series (Tag D-W2QAA)
Let \(G\) be a group. Set \(G^{(0)} \coloneqq G\) and \(G^{(i+1)} \coloneqq [G^{(i)}, G^{(i)}]\), the commutator subgroup of \(G^{(i)}\), for \(i\geq 0\). The derived series of \(G\) is the chain \(G^{(0)} \geq G^{(1)} \geq G^{(2)} \geq \cdots\).
(Implications among the classes)
\[ \text{cyclic} \implies \text{abelian} \implies \text{nilpotent} \implies \text{solvable}, \] and no implication reverses: \(\ZZ/2\times\ZZ/2\) is abelian and not cyclic, \(Q_8\) is nilpotent and not abelian, and \(S_3\) is solvable and not nilpotent. The symmetric group \(S_n\) is solvable if and only if \(n\leq 4\). Over a field of characteristic \(0\), a polynomial is solvable by radicals if and only if its Galois group is solvable; see Cyclotomic and radical extensions.
Nilpotent groups
Nilpotent group (Tag D-53JVH)
A group \(G\) is nilpotent if its upper central series \(Z_0\leq Z_1\leq\cdots\) satisfies \(Z_n = G\) for some \(n\geq 0\).
Nilpotent groups via the upper central series (Tag T-7PU33)
Let \(G\) be a group with upper central series \(Z_0\leq Z_1\leq\cdots\). Then \(G\) is nilpotent if and only if \(Z_n=G\) for some \(n\geq 0\).
Nilpotent groups via the lower central series (Tag T-4INET)
Let \(G\) be a group with identity \(e\) and lower central series \(G^0\geq G^1\geq\cdots\). Then \(G\) is nilpotent if and only if \(G^n=\theset{e}\) for some \(n\geq0\).
A finite group is nilpotent if and only if its Sylow subgroups are normal (Tag T-GM7EB)
Let \(G\) be a finite group. Then \(G\) is nilpotent if and only if, for every prime \(p\) dividing \(\abs{G}\), the Sylow $p$-subgroup of \(G\) is normal.
A finite group is nilpotent if and only if its maximal subgroups are normal (Tag T-OHEFT)
A finite group \(G\) is nilpotent if and only if every maximal subgroup of \(G\) is normal in \(G\).
For a finite group \(G\), the following are equivalent.
\(G\) is nilpotent.
For every proper subgroup \(H < G\), \(H < N_G(H)\).
Every Sylow subgroup of \(G\) is normal.
\(G\) is the direct product of its Sylow subgroups.
Every maximal subgroup of \(G\) is normal.
The lower central series of \(G\) reaches \(1\).
The upper central series of \(G\) reaches \(G\).
Subgroups and quotients of nilpotent groups are nilpotent.
An extension of nilpotent groups need not be nilpotent: \(S_3\) has the normal subgroup \(A_3 \cong C_3\) with quotient \(S_3/A_3 \cong C_2\), both nilpotent, and \(S_3\) is not nilpotent.
For \(N \normal G\), \(G\) is solvable if and only if both \(N\) and \(G/N\) are solvable.
A finite nilpotent group \(G\) has a normal subgroup of order \(d\) for every \(d\) dividing \(\abs{G}\).