• The product of two or more commutators need not be a commutator. It is known that the least order of a finite group for which there exists two commutators whose product is not a commutator is 96.
• Note that the inverse [x, y]−1 of [x, y] is given by [x, y]−1 = [y, x] =
yxy−1x−1.
Here are a list of properties of the commutator subgroupG′.
Lemma 1.46. Let G′ be the commutator subgroup ofG. 1. G′ is characteristic inG.
2. Gis abelian if and only ifG′ is trivial. 3. G/G′ is abelian.
4. IfN is normal in G, thenG/N is abelian if and only ifG′≤N.
Proof. 1. To show that G′ is characteristic in G, we have to show that f(G′) =G′ forf any automorphism ofG. Now
f([x, y]) =f(xyx−1y−1) =f(x)f(y)f(x)−1f(y)−1= [f(x), f(y)].
2. We have thatG′is trivial if and only ifxyx−1y−1= 1 which exactly means
thatxy=yx.
3. Since G′ is characteristic, it is also normal in G, and G/G′ is a group. We are left to prove it is an abelian group. Take two elements (that is two cosets)G′xand G′y inG/G′. We have thatG′xG′y =G′yG′x ⇐⇒ G′xy=G′yxby definition of the law group onG/G′. Now
G′xy=G′yx ⇐⇒ xy(yx)−1∈G′ ⇐⇒ xyx−1y−1∈G′, which holds by definition.
4. Let us assume that N is normal inG. We have thatG/N is a group, and G/N is abelian if and only if forN x,N ytwo cosets we have
N xN y=N yN x ⇐⇒ N xy=N yx ⇐⇒ xy(yx)−1∈N ⇐⇒ xyx−1y−1∈N which exactly tells that each commutator must be inN.
We can iterate the process of taking commutators:
G(0)=G, G(1)=G′, G(2)= (G′)′, . . . , G(i+1)= (G(i))′, . . .
Definition 1.34. The group G is said to be solvable if G(r) = 1 for some r. We then have anormal series
{1}=G(r)EG(r−1)E· · ·EG(0)=G
called thederived series ofG. The term“solvable” historically refers to Galois theory and the question of “solvability” of quintic equations, as we will see later. We have already seen the notion of subnormal series in the previous section. By normal series, we mean a serie where not only each group is normal in its successor, but also each group is normal in the whole group, namely each G(i)
is normal inG. We have indeed such series here using the fact that the commu- tator subgroup is a characteristic subgroup, which is furthermore a transitivity property.
Let us make a few remarks about the definition of solvable group.
Lemma 1.47. 1. Every abelian group is solvable.
2. A groupGboth simple and solvable is cyclic of prime order. 3. A non-abelian simple groupGcannot be solvable.
Proof. 1. We know thatGis abelian if and only ifG′ is trivial. We thus get the normal series
G(0) =G⊲G(1)={1}.
2. IfGis simple, then its only normal subgroups are{1} andG. SinceG′ is characteristic and thus normal, we have eitherG′ ={1}or G′=G. The latter cannot possibly happen, since then the derived serie cannot reach {1}which contradicts the fact thatGis solvable. Thus we must have that G′ ={1}, which means that Gis abelian. We conclude by remembering that an abelian simple group must be cyclic of order a primep.
3. If Gis non-abelian, thenG′ cannot be trivial, thus sinceGis simple, its only normal subgroups can be either{1}or {G}, thusG′ must be either one of the other, and it cannot be{1}, so it must beG. Thus the derived series never reaches{1} andGcannot be solvable.
There are several ways to define solvability.
Proposition 1.48. The following conditions are equivalent. 1. Gis solvable, that is, it has a derived series
{1}=G(r)EG(r−1)E· · ·EG(0)=G. 2. Ghas a normal series
{1}=GrEGr−1E· · ·EG0=G
1.11. SOLVABLE AND NILPOTENT GROUPS 61
3. Ghas a subnormal series
{1}=GrEGr−1E· · ·EG0=G
where all factors, that is all quotient groupsGi/Gi+1 are abelian.
Proof. That 1.⇒ 2. is clear from Lemma 1.46 where we proved that G/G′ is abelian, whereG′ is the commutator subgroup ofG.
That 2.⇒3. is also clear since the notion of normal series is stronger than subnormal series.
What we need to prove is thus that 3.⇒1.Starting fromG, we can always compute G′, then G(2), . . ..To prove that G has a derived series, we need to
check thatG(s)={1} for somes. Suppose thus thatGhas a subnormal series
1 =GrEGr−1E· · ·EG0=G
where all quotient groups Gi/Gi+1 are abelian. For i= 0, we getG1EGand
G/G1 is abelian. By Lemma 1.46, we know thatG/G1 is abelian is equivalent
to G′ ≤ G
1. By induction, let us assume that G(i) ≤ Gi, that is taking i
times the derived subgroup of G is a subgroup which is contained in the ith termGi of the subnormal series, and see what happens with G(i+1). We have
that G(i+1) = (G(i))′≤G′
i by induction hypothesis (and noting that if H ⊂G
then H′ ⊂ G′, since all the commutators in H surely belong to those of G). Furthermore,G′
i≤Gi+1 sinceGi/Gi+1 is abelian. ThusG(r)≤Gr={1}.
Let us see what are the properties of subgroups and quotients of solvable groups.
Proposition 1.49. Subgroups and quotients of a solvable group are solvable. Proof. Let us first consider subgroups of a solvable groups. IfH is a subgroup of a solvable group G, then H is solvable because H(i) ≤ G(i) for all i, and
in particular for r such thatH(r)≤G(r) ={1} which proves that the derived
series ofH terminates.
Now considerNa normal subgroup of a solvable groupG. The commutators of G/N are cosets of the form xN yN x−1N y−1N = xyx−1y−1N, so that the commutator subgroup (G/N)′ of G/N satisfies (G/N)′ =G′N/N (we cannot write G′/N since there is no reason for N to be a subgroup of G′). Induc- tively, we have (G/N)(i)=G(i)N/N. SinceGis solvable,G(r)={1} and thus
(G/N)(r)=N/N ={1} which shows thatG/N is solvable.
Example 1.34. Consider the symmetric groupS4. It has a subnormal series
{1}⊳C2×C2⊳A4⊳S4,
where A4 is the alternating group of order 12 (given by the even permutations
of 4 elements) and C2×C2is the Klein group of order 4 (corresponding to the
permutations 1,(12)(34),(13)(24),(14)(23)). The quotient groups are C2×C2/{1} ≃C2×C2 abelian of order 4
A4/C2×C2 ≃C3 abelian of order 3
We finish by introducing the notion of a nilpotent group. We will skip the general definition, and consider only finite nilpotent groups, for which the following characterization is available.
Proposition 1.50. The following statements are equivalent. 1. Gis the direct product of its Sylow subgroups.
2. Every Sylow subgroup of Gis normal.
Proof. IfGis the direct product of its Sylow subgroups, that every Sylow sub- group of G is normal is immediate since the factors of a direct product are normal subgroups.
Assume that every Sylow subgroup ofG is normal, then by Lemma 1.37, we know that every normal Sylow p-subgroup is unique, thus there is a unique Sylow pi-subgroup Pi for each prime divisor pi of |G|, i = 1, . . . , k. Now by
Lemma 1.32, we have that |P1P2| = |P1||P2| since P1 ∩P2 = {1}, and thus
|P1· · ·Pk|=|P1| · · · |Pk|=|G|by definition of Sylow subgroups. Since we work
with finite groups, we deduce thatG is indeed the direct product of its Sylow subgroups, having that G=P1· · ·Pk andPi∩Qj6=iPj is trivial.
Definition 1.35. A finite groupGwhich is the product of its Sylow subgroups, or equivalently by the above proposition satisfies that each of its Sylow subgroup is normal is called anilpotent group.
Corollary 1.51. Every finite abelian group and every finitep-group is nilpotent. Proof. A finite abelian group surely has the property that each of its Sylow subgroup is normal, so it is nilpotent.
Now considerP a finitep-group. Then by definition P has only one Sylow subgroup, namely itself, so it is the direct product if its Sylow subgroups and thus is nilpotent.
Finite nilpotent groups are also nicely described with respect to their nor- malizer.
Proposition 1.52. If Gis a finite nilpotent group, then no proper subgroupH of Gis equal to its normalizerNG(H) ={g∈G, gH=Hg}.
Proof. Let H be a proper subgroup ofG, and let nbe the largest index such that Gn ⊆H (such index exists sinceG is nilpotent). There existsa∈ Gn+1
such that a 6∈ G (since H is a proper subgroup). Now for every h ∈ H, the cosets aGn andhGn commute (sinceGn+1/Gn⊆Z(G/Gn)), namely:
Gnah= (Gna)(Gnh) = (Gnh)(Gna) =Gnha
and thus there is someh′ ∈Gn⊆H for which ah=h′ha that is
aha−1=h′h∈H. Thusa∈NG(H) anda6∈H.
1.11. SOLVABLE AND NILPOTENT GROUPS 63
Here is the definition for possibly infinite nilpotent groups.
Definition 1.36. Acentral seriesfor a groupGis a normal series
{1}=GnEGn−1E· · ·EG0=G
such thatGi/Gi+1⊆Z(G/Gi+1) for everyi= 0, . . . , n−1. An arbitrary group
Gis said to benilpotent if it has a central series. The smallestnsuch that G has a central series of lengthnis called thenilpotency classofG, andGis said to be nilpotent of classn.
Example 1.35. Abelian groups are nilpotent of class 1, since
{1}=G1EG0=G
is a normal series forGand fori= 0 we haveG/{1} ≃G⊆Z(G).
Nilpotent groups in general are discussed with solvable groups since they can be described with normal series, and one can prove that they are solvable. Indeed, ifGi/Gi+1⊆Z(G/Gi+1), then the elements ofGi/Gi+1 commute with
each other, since they commute with everything in G/Gi+1, thus Gi/Gi+1 is
abelian. It is not true that solvable groups are necessarily nilpotent (see Exer- cises for an example).
The main definitions and results of this chapter are • (1.1-1.2). Definitions of: group, subgroup, group
homomorphism, order of a group, order of an element, cyclic group.
• (1.3-1.4). Lagrange’s Theorem. Definitions of: coset, normal subgroup, quotient group
• (1.5). 1st, 2nd and 3rd Isomorphism Theorems. • (1.6). Definitions of: external (semi-)direct product,
internal (semi-)direct product.
• (1.7). Cayley’s Theorem, the Orbit-Stabilizer The- orem, the Orbit-Counting Theorem. Definitions of: symmetric group, group action, orbit, transitive ac- tion, stabilizer, centralizer. That the orbits partition the set under the action of a group
• (1.8). Definition: p-group, Sylowp-subgroup. The 3 Sylow Theorems, Cauchy Theorem
• (1.9). Definition: simple group. Applications of the Sylow Theorems.
• (1.10). Definitions: subnormal series, composition series. Jordan-H¨older Theorem.
• (1.11). Definitions: characteristic subgroup, commu- tator subgroup, normal and derived series, solvable group, finite nilpotent group.