Solvable and Nilpotent Groups

Introduction

The commutator $[x,y] = x^{-1}y^{-1}xy$ measures the failure of $x$ and $y$ to commute, and the subgroups it generates measure the failure of a group to be abelian. Iterating the construction produces two descending chains, the derived series and the lower central series, whose termination defines the solvable and the nilpotent groups. These two classes are the best-behaved infinite families in group theory: they are closed under subgroups and quotients, they carry a canonical filtration whose successive quotients are abelian, and in the finite case they are governed by strong structural theorems. Nilpotent groups form a proper subclass of the solvable ones, and the finite nilpotent groups are exactly the direct products of their Sylow subgroups.

This article is the eleventh of the corpus and the second of the group articles, below Infinite Abelian Groups and beside the Groups, Group Actions and Structure. It uses the elementary theory of Groups — subgroups, normal subgroups, quotients, the isomorphism theorems, the centre, the Sylow theorems — the direct products, and the abelian structure theory of Infinite Abelian Groups, whose additive notation it follows for abelian groups. It supplies the structural background, for the study of infinite groups, and for the solvability criterion used in the classification of finite simple groups.

The article is purely algebraic. The classical examples of nilpotent groups drawn from unitriangular matrices belong to the linear theory, which requires the modules and vector spaces of the later Part I articles, and they are named and deferred; the examples proved here are the permutation groups and the finite groups of small order, which the earlier group articles provide. The geometric material — the Cayley graph, the growth of a group, the reflection representation of a Coxeter group — belongs to Part II and is not used.

Commutators and the Derived Series

Commutators and the Commutator Subgroup

Definition. Let $G$ be a group. The commutator of $x, y \in G$ is $[x,y] = x^{-1}y^{-1}xy$, and for subgroups $A, B \leq G$ the commutator subgroup $[A,B]$ is the subgroup generated by all commutators $[a,b]$ with $a \in A$, $b \in B$.

Proposition. The following identities hold in every group.

$[x,y]^{-1} = [y,x]$.
$[x,yz] = [x,z]\, z^{-1}[x,y]z$ and $[xy,z] = y^{-1}[x,z]y\,[y,z]$.
$[x,y] = 1$ if and only if $x$ and $y$ commute.
If $N \trianglelefteq G$ then $G/N$ is abelian if and only if $G' \leq N$; in particular $G'$ is the least normal subgroup with abelian quotient. Also $N \leq Z(G)$ if and only if $[G,N] = 1$.

Proof. (1) and (3) are immediate from the definition. (2) Expanding $[x,yz] = x^{-1}z^{-1}y^{-1}xyz$ and regrouping the middle gives $x^{-1}z^{-1}xz \cdot z^{-1}x^{-1}y^{-1}xyz = [x,z]\cdot z^{-1}[x,y]z$; the second identity is verified by the same expansion of $[xy,z] = y^{-1}x^{-1}z^{-1}xyz$ as $y^{-1}x^{-1}z^{-1}xz\,y \cdot y^{-1}z^{-1}yz = y^{-1}[x,z]y\cdot[y,z]$. (4) The quotient $G/N$ is abelian if and only if $xyN = yxN$ for all $x,y$, that is, $[x,y] \in N$ for all $x,y$, which says $G' \leq N$; the case $N = G'$ shows that $G/G'$ is abelian, so $G'$ is the least such normal subgroup. Finally $N \leq Z(G)$ says every $n \in N$ commutes with every $g \in G$, which is $[g,n] = 1$ for all such $g,n$, that is, $[G,N] = 1$.

The quotient $G/[G,G]$ is the abelianisation of $G$, and it is the largest abelian quotient of $G$: every homomorphism from $G$ to an abelian group factors uniquely through it. This is the universal property of the abelianisation, and it is the first instance of the pattern of the next section.

The Derived Series

Definition. The derived series of a group $G$ is the descending chain

$$ G^{(0)} = G, \qquad G^{(i+1)} = [G^{(i)}, G^{(i)}], \qquad G^{(0)} \trianglerighteq G^{(1)} \trianglerighteq G^{(2)} \trianglerighteq \cdots $$

Each $G^{(i)}$ is characteristic in $G$, hence normal, and each quotient $G^{(i)}/G^{(i+1)}$ is abelian.

Definition. A group $G$ is solvable if $G^{(n)} = 1$ for some $n \geq 0$. The least such $n$ is the derived length of $G$; thus the groups of derived length $0$ are trivial, those of derived length at most $1$ are abelian, and those of derived length $2$ are the nonabelian groups with abelian derived subgroup.

Proposition. Let $N \trianglelefteq G$.

If $G$ is solvable then every subgroup and every quotient of $G$ is solvable, and $\operatorname{dl}(H) \leq \operatorname{dl}(G)$ for subgroups and quotients.
If $N$ and $G/N$ are solvable then $G$ is solvable, with $\operatorname{dl}(G) \leq \operatorname{dl}(N) + \operatorname{dl}(G/N)$.
If $G$ is solvable then $G' = G^{(1)} \neq G$ unless $G$ is trivial; a nontrivial solvable group has a nontrivial abelian quotient.

Proof. (1) The derived series of a subgroup satisfies $H^{(i)} \leq G^{(i)}$, by induction on $i$ from the definition of the commutator subgroup; for a quotient, $(G/N)^{(i)} = G^{(i)}N/N$. (2) Let $m = \operatorname{dl}(G/N)$ and $n = \operatorname{dl}(N)$. Then $G^{(m)} \leq N$, so $G^{(m+n)} \leq N^{(n)} = 1$. (3) If $G^{(1)} = G$ then the derived series is constant and never reaches $1$; the abelianisation $G/G'$ is then nontrivial when $G \neq 1$.

Corollary. A group $G$ is solvable if and only if it has a subnormal series — a chain $1 = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$ — whose successive quotients $G_{i+1}/G_i$ are abelian.

Proof. Given a subnormal series with abelian quotients, each quotient $G_{i+1}/G_i$ being abelian means $G_{i+1}' \leq G_i$, so the derived series of $G$ descends at least one step per term of the series and reaches $1$. Conversely the derived series itself is such a series, because each $G^{(i)}/G^{(i+1)}$ is abelian.

The corollary is the definition of solvability in terms of a filtration by abelian layers, and it is the form in which solvability is usually verified: it suffices to exhibit one subnormal series with abelian factors.

Solvable Groups: Examples

Example (abelian groups). Every abelian group is solvable of derived length at most $1$. The additive group $\mathbb{Z}$ and the cyclic groups $\mathbb{Z}/n\mathbb{Z}$ of Infinite Abelian Groups are examples, and every quotient of a solvable group is solvable.

Example (the symmetric group $S_3$). The derived subgroup of $S_3$ is $A_3 \cong \mathbb{Z}/3\mathbb{Z}$, which is abelian, and $S_3/A_3 \cong \mathbb{Z}/2\mathbb{Z}$; hence $S_3^{(2)} = 1$ and $S_3$ is solvable of derived length $2$. Its centre is trivial, so it is not nilpotent, as the next section shows.

Example (the symmetric group $S_4$). The derived subgroup of $S_4$ is $A_4$, and the derived subgroup of $A_4$ is the Klein four-group $V_4 = \{e, (12)(34), (13)(24), (14)(23)\}$, which is abelian. Hence $S_4^{(3)} = 1$: the symmetric group on four letters is solvable of derived length $3$. The derived series $S_4 \trianglerighteq A_4 \trianglerighteq V_4 \trianglerighteq 1$ is computed in the verification accompanying this article.

Example (the alternating group $A_5$). The group $A_5$ is simple and nonabelian, so its only normal subgroups are $1$ and $A_5$; since $A_5' \trianglelefteq A_5$ and $A_5$ is not abelian, $A_5' = A_5$, and the derived series is constant. Thus $A_5$ is not solvable, and no group containing it as a subgroup or quotient — in particular no symmetric group $S_n$ with $n \geq 5$ — is solvable, because solvability is inherited by subgroups and quotients.

The last example is the reason solvability is the dividing line in the theory of equations and in the classification of finite simple groups: the nonsolvability of $A_5$ for $n \geq 5$ is the group-theoretic content of the unsolvability of the quintic by radicals, and the classification of finite simple groups is a classification of the obstacles to solvability.

Nilpotent Groups

The Lower Central Series

Definition. The lower central series of a group $G$ is

$$ \gamma_1 G = G, \qquad \gamma_{n+1} G = [\gamma_n G, G], $$

so that $\gamma_2 G = G'$ and $\gamma_1 G \geq \gamma_2 G \geq \gamma_3 G \geq \cdots$ is a descending chain of characteristic subgroups with $\gamma_{n}G/\gamma_{n+1}G$ central in $G/\gamma_{n+1}G$.

Definition. A group $G$ is nilpotent if $\gamma_{c+1}G = 1$ for some $c$; the least such $c$ is the nilpotency class of $G$. Nilpotent groups of class $0$ are trivial, of class at most $1$ are abelian, and of class $2$ satisfy $[G,G] \leq Z(G)$.

Proposition. If $G$ is nilpotent of class $c$ then every subgroup and every quotient of $G$ is nilpotent of class at most $c$, and the centre of a nontrivial nilpotent group is nontrivial.

Proof sketch. For subgroups, $\gamma_n H \leq \gamma_n G$ by induction. For quotients, $\gamma_n(G/N) = \gamma_n G \cdot N/N$. For the centre, if $\gamma_c G \neq 1$ and $\gamma_{c+1}G = 1$ then $\gamma_c G$ is central in $G$, because $[\gamma_c G, G] = \gamma_{c+1}G = 1$.

The last statement is the key structural fact about nilpotent groups: the descending central series reaches the centre from the bottom, so a nontrivial nilpotent group has a nontrivial centre and one can argue by induction on the class by passing to the quotient by the centre.

The Upper Central Series

Definition. The upper central series of $G$ is the ascending chain of characteristic subgroups

$$ Z_0(G) = 1, \qquad Z_{i+1}(G)/Z_i(G) = Z(G/Z_i(G)), $$

so that $Z_1(G) = Z(G)$ and $Z_i(G) \trianglelefteq G$ for all $i$, and $Z_i(G) \leq Z_{i+1}(G)$.

Definition. A group $G$ is nilpotent of class at most $c$ if $Z_c(G) = G$.

Theorem. The following are equivalent for a group $G$:

$G$ is nilpotent of class at most $c$, that is, $\gamma_{c+1}G = 1$;
$Z_c(G) = G$;
there is a central series $1 = N_0 \leq N_1 \leq \cdots \leq N_c = G$ with $N_i \trianglelefteq G$ and $N_{i+1}/N_i \leq Z(G/N_i)$ for all $i$.

Proof sketch. (1) $\Rightarrow$ (3): the lower central series is such a series, because $\gamma_n G/\gamma_{n+1}G$ is central in $G/\gamma_{n+1}G$; reindexing from the bottom gives the required chain. (3) $\Rightarrow$ (2): prove $N_{c-i} \leq Z_i(G)$ by induction on $i$, so that $G = N_c \leq Z_c(G)$. (2) $\Rightarrow$ (1): prove $\gamma_{i+1}G \leq Z_{c-i}(G)$ by induction on $i$, so that $\gamma_{c+1}G \leq Z_0(G) = 1$.

Corollary. Every nilpotent group is solvable, and $\operatorname{dl}(G) \leq \operatorname{cl}(G)$. The converse fails: $S_3$ is solvable and has trivial centre, hence is not nilpotent.

Example (finite $p$-groups). Every finite $p$-group is nilpotent: a nontrivial finite $p$-group has a nontrivial centre by the class equation, as proved in Groups, and the quotient by the centre is again a $p$-group of smaller order, so induction on the order shows that the upper central series reaches $G$. The quotients $\gamma_iG/\gamma_{i+1}G$ are elementary abelian $p$-groups.

Example (the quaternion and dihedral groups). The quaternion group $Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}$ has centre $\{\pm 1\}$, derived subgroup $\{\pm 1\} = \gamma_2Q_8$ and $Q_8/\{\pm1\} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ abelian; hence $\gamma_3Q_8 = 1$ and $Q_8$ is nilpotent of class $2$. The dihedral group $D_8 = \langle r, s \mid r^4 = s^2 = 1,\ srs = r^{-1}\rangle$ has centre $\{1, r^2\}$ and the same argument gives nilpotency of class $2$. Both computations are reproduced in the verification accompanying this article.

Theorems on Nilpotent and Solvable Groups

Hall Subgroups

Definition. Let $\pi$ be a set of primes. A $\pi$-group is a group all of whose element orders have prime factors in $\pi$, a $\pi$-number is an integer all of whose prime factors lie in $\pi$, and a Hall $\pi$-subgroup of a finite group $G$ is a $\pi$-subgroup $H \leq G$ such that the index $[G:H]$ is a $\pi'$-number, where $\pi'$ is the complement of $\pi$ in the primes. A Hall $\{p\}$-subgroup is a Sylow $p$-subgroup.

Theorem (Hall). Let $G$ be a finite solvable group and let $\pi$ be a set of primes.

$G$ has a Hall $\pi$-subgroup.
Any two Hall $\pi$-subgroups of $G$ are conjugate.
Every $\pi$-subgroup of $G$ is contained in a Hall $\pi$-subgroup.

Conversely, if a finite group has a Hall $\pi$-subgroup for every set of primes $\pi$, then it is solvable.

Proof sketch. The proof is by induction on $|G|$, using the existence of a nontrivial abelian normal subgroup — supplied by the derived series, whose last nontrivial term is abelian and normal — and reducing to the quotient by a minimal normal subgroup. Since solvable groups have composition factors of prime order, the reduction step splits into the cases of a normal subgroup of prime index and of a normal elementary abelian subgroup, and in each case the Hall subgroup of the quotient is lifted along a Sylow argument. The converse is proved by exhibiting a set of primes for which no Hall subgroup exists in a nonsolvable group, using that a nonsolvable group has a composition factor isomorphic to a nonabelian simple group, in which two distinct maximal subgroups supply the obstruction; the details are in the references.

For $p$-groups, the Hall subgroups specialise to the Sylow theorems: a finite $p$-group is its own Sylow $p$-subgroup, and the Sylow theorems of Group Actions and Structure give existence, conjugacy and containment for the prime case.

The Fitting and Frattini Subgroups

Definition. The Fitting subgroup $F(G)$ of a finite group $G$ is the product of all normal nilpotent subgroups of $G$; the product of finitely many normal nilpotent subgroups is normal and nilpotent, so $F(G)$ is the largest normal nilpotent subgroup of $G$. The Frattini subgroup $\Phi(G)$ is the intersection of all maximal subgroups of $G$, with $\Phi(G) = G$ when $G$ has no maximal subgroup.

Proposition. Let $G$ be a finite group.

$F(G)$ is characteristic in $G$ and contains the centre of $G$; if $G$ is nilpotent then $F(G) = G$.
$\Phi(G)$ is characteristic in $G$, and $\Phi(G)$ is nilpotent; in fact $\Phi(G) \leq F(G)$.
A subgroup $H \leq G$ satisfies $H\Phi(G) = G$ only if $H = G$ (the non-generator property), and for a finite $p$-group $\Phi(G) = G^p[G,G]$, where $G^p$ is the subgroup generated by the $p$-th powers.

Proof sketch. (1) The product of normal nilpotent subgroups is normal, and the product of two normal nilpotent subgroups is nilpotent by Fitting's theorem; the centre of $G$ is a normal nilpotent subgroup, so it lies in $F(G)$; and $G$ itself is normal and nilpotent when $G$ is nilpotent. (2) Characteristic is immediate from the definition; $\Phi(G)$ is nilpotent because a finite group all of whose maximal subgroups are normal with prime-power index is nilpotent, and the Frattini argument shows that $\Phi(G)$ is contained in the Fitting subgroup. For (3), the non-generator property follows from the definition, and the formula for a $p$-group is the Burnside basis theorem: $G/\Phi(G)$ is elementary abelian of order $p^{d}$ with $d$ the minimal number of generators, and $G^p[G,G]$ is the least normal subgroup with that quotient.

Theorem. For a finite group $G$, the following are equivalent: (i) $G$ is nilpotent; (ii) every Sylow subgroup of $G$ is normal; (iii) $G$ is the direct product of its Sylow subgroups; (iv) every maximal subgroup of $G$ is normal.

Proof sketch. (i) $\Rightarrow$ (ii): a Sylow $p$-subgroup $P$ of a nilpotent group satisfies $N_G(P) = G$, because if $N_G(P) \neq G$ then $N_G(P) \neq N_G(N_G(P))$ in a nilpotent group, and this contradicts the maximality of $P$ among its conjugates. (ii) $\Rightarrow$ (iii): distinct Sylow subgroups of coprime order intersect trivially and commute when both are normal, so the product map is an isomorphism. (iii) $\Rightarrow$ (i): a direct product of $p$-groups is nilpotent. (i) $\Leftrightarrow$ (iv): a maximal subgroup of a nilpotent group is normal because it contains the normaliser of its own normaliser, and conversely a group all of whose maximal subgroups are normal is nilpotent.

The equivalence (i) $\Leftrightarrow$ (iii) is the structure theorem for finite nilpotent groups: they are exactly the direct products of their Sylow subgroups. It gives a second proof that finite $p$-groups are nilpotent, and it shows that every finite nilpotent group is the direct product of its primary components, in the additive language of Infinite Abelian Groups.

Solvability Criteria

Theorem (Burnside). Every finite group of order $p^a q^b$, with $p$ and $q$ primes, is solvable.

Theorem (Feit–Thompson). Every finite group of odd order is solvable.

The first is proved by a character-theoretic argument on the vanishing of characters on conjugacy classes; the second is the deep theorem that the solvable groups of odd order exhaust the odd-order finite groups, and it is the first step of the classification of finite simple groups, where it rules out all odd-order nonabelian simple groups. Both are stated here without proof; the second is used.

Theorem (Wielandt). A finite group is nilpotent if and only if every maximal subgroup is normal; equivalently, the nilpotent groups are the finite groups all of whose subgroups are subnormal.

Theorem (the derived subgroup of a nilpotent group). If $G$ is nilpotent of class $c$ then $G'$ is nilpotent of class at most $c - 1$ and $\gamma_i G' \leq \gamma_{i+1}G$; in particular a group of class $2$ has abelian derived subgroup.

Proof sketch. By induction on $i$: $\gamma_1 G' = G' = \gamma_2 G$, and $\gamma_{i+1}G' = [\gamma_i G', G'] \leq [\gamma_{i+1}G, G] = \gamma_{i+2}G$. At $i = c-1$ this gives $\gamma_c G' \leq \gamma_{c+1}G = 1$.

Examples from Permutation Groups

Symmetric and Alternating Groups

The derived series of the symmetric groups is the source of the standard examples, and the computations are elementary once the commutator subgroup of $S_n$ is known.

Theorem. For $n \geq 2$, the derived subgroup of $S_n$ is $A_n$; for $n \geq 5$, the derived subgroup of $A_n$ is $A_n$. Consequently $S_n$ is solvable for $n \leq 4$ and nonsolvable for $n \geq 5$.

Proof sketch. The quotient $S_n/A_n \cong \mathbb{Z}/2\mathbb{Z}$ is abelian, so $S_n' \leq A_n$; the transposition identity $[(ab),(bc)] = (abc)$ and its conjugates show that $S_n'$ contains a $3$-cycle, and the $3$-cycles generate $A_n$, so $S_n' = A_n$. For $n \geq 5$, that $A_n$ is nonabelian and simple gives $A_n' = A_n$.

Example. The solvable cases are $S_2 \cong \mathbb{Z}/2$, $S_3$ of derived length $2$, and $S_4$ of derived length $3$, with derived series

$$ S_4 \trianglerighteq A_4 \trianglerighteq V_4 \trianglerighteq 1. $$

The case $S_4$ is the largest solvable symmetric group, and the constant derived series of $A_5$ is the smallest obstruction; this dichotomy is the reason solvable groups are the natural setting for Galois theory.

Definition. The dihedral group $D_{2n}$ of order $2n$ is $\langle r, s \mid r^n = s^2 = 1,\ srs = r^{-1}\rangle$, the symmetry group of a regular $n$-gon as an abstract group.

Proposition. For $n \geq 2$ the group $D_{2n}$ is solvable of derived length at most $2$: its derived subgroup is $\langle r^2\rangle$, of index at most $2$ in the cyclic group $\langle r\rangle$ and abelian.

Proof. The quotient $D_{2n}/\langle r\rangle \cong \mathbb{Z}/2$ is abelian, so $D_{2n}' \leq \langle r\rangle$. The commutator $[r,s] = r^{-1}s^{-1}rs = r^{-2}$ generates $\langle r^2\rangle$, and $\langle r^2\rangle$ is normal in $D_{2n}$ with $D_{2n}/\langle r^2\rangle$ abelian — of order $4$ when $n$ is even and of order $2$ when $n$ is odd — so $D_{2n}' = \langle r^2\rangle$, which is cyclic and hence abelian.

Proposition. $D_{2n}$ is nilpotent if and only if $n$ is a power of $2$; in that case it is a $2$-group.

Proof sketch. If $n = 2^k$ then $|D_{2n}| = 2^{k+1}$ and $D_{2n}$ is a $2$-group, hence nilpotent. Otherwise write $n = 2^k m$ with $m > 1$ odd. The Sylow $q$-subgroups for the odd primes $q \mid n$ are characteristic in the cyclic normal subgroup $\langle r\rangle$, hence normal; but the Sylow $2$-subgroup $P = \langle r^m, s\rangle$, of order $2^{k+1}$, has the $m$ distinct conjugates $\langle r^m, r^{2i}s\rangle$ for $i = 0, 1, \ldots, m-1$, so it is not normal, and the case $k = 0$ is included. By the equivalence of nilpotency with the normality of every Sylow subgroup, $D_{2n}$ is not nilpotent.

The example $D_8$, the case $n = 4$, is the smallest nonabelian nilpotent dihedral group, and $D_{2p}$ for an odd prime $p$ is the smallest nonabelian group that is solvable but not nilpotent, since its $p$ Sylow $2$-subgroups, each of order $2$, are not normal.

Remark. The group of upper unitriangular matrices over a commutative ring, the group of upper triangular matrices with all diagonal entries equal to $1$, is nilpotent of class $n-1$; the linear algebra of matrices and vector spaces is developed, and the example is quoted there. The Heisenberg group over $\mathbb{Z}$, the same construction in dimension $3$, is the standard nonabelian nilpotent group of class $2$ that is not a $p$-group.

Summary

The commutator $[x,y] = x^{-1}y^{-1}xy$ generates the commutator subgroup $[A,B]$, and $G' = [G,G]$ is the least normal subgroup with abelian quotient; $G/G'$ is the abelianisation, the largest abelian quotient. The derived series $G^{(i)}$ has abelian factors, and $G$ is solvable when the series reaches $1$; solvability is inherited by subgroups and quotients, is preserved by extensions, and is equivalent to the existence of a subnormal series with abelian factors. Abelian groups are solvable of length at most $1$; $S_3$ and $S_4$ are solvable, of lengths $2$ and $3$; $A_5$ and all $S_n$ with $n \geq 5$ are not.

The lower central series $\gamma_n G = [\gamma_{n-1}G, G]$ and the upper central series $Z_i(G)$ define nilpotency: $G$ is nilpotent of class at most $c$ when $\gamma_{c+1}G = 1$, equivalently when $Z_c(G) = G$, equivalently when $G$ admits a finite central series. Nilpotent groups are solvable, and the converse fails for $S_3$; finite $p$-groups and the groups $Q_8$ and $D_8$ are nilpotent of class $2$. The finite nilpotent groups are exactly those that are the direct product of their Sylow subgroups, equivalently those in which every Sylow subgroup, or every maximal subgroup, is normal.

Hall's theorem gives the existence, conjugacy and containment of Hall $\pi$-subgroups in a finite solvable group, and their existence for all $\pi$ characterises solvability; the Sylow theorems are the case of a single prime. The Fitting subgroup $F(G)$ is the largest normal nilpotent subgroup and the Frattini subgroup $\Phi(G)$ is the intersection of the maximal subgroups, which is nilpotent and contained in $F(G)$ and, for a $p$-group, equals $G^p[G,G]$. Burnside's theorem makes every group of order $p^aq^b$ solvable and the Feit–Thompson theorem makes every group of odd order solvable. The permutation groups provide the examples: the derived series of $S_n$ and $A_n$, and the dihedral groups, solvable always and nilpotent exactly when the order is a power of $2$.

Summary of Notation

Symbol	Meaning
$[x,y]$, $[A,B]$	Commutator of elements; commutator subgroup of two subgroups
$G' = [G,G]$	Derived subgroup; least normal subgroup with abelian quotient
$G^{(i)}$	$i$-th term of the derived series
$\operatorname{dl}(G)$	Derived length
$\gamma_n G$	$n$-th term of the lower central series
$Z(G)$, $Z_i(G)$	Centre; $i$-th term of the upper central series
$\operatorname{cl}(G)$	Nilpotency class
$F(G)$	Fitting subgroup: largest normal nilpotent subgroup
$\Phi(G)$	Frattini subgroup: intersection of maximal subgroups
$G^p$	Subgroup generated by the $p$-th powers
$A_n$, $S_n$, $V_4$	Alternating, symmetric groups; Klein four-group
$Q_8$, $D_{2n}$	Quaternion group; dihedral group of order $2n$
$\pi$, $\pi'$	Set of primes; its complement
$[G:H]$	Index of a subgroup