3.5. Procesamiento y análisis de datos
3.5.1. Descripción del sistema a simular
Let π be a set of primes. Recall that a positive numbern is a π-number if the prime divisors of n are all in π; or equivalently, the π-part nπ of n is
equal ton.
Let G be a finite group. An element x in G is called a π-element if its order o(x) is a π-number; and Gis called aπ-group if|G|π =|G|.
Since |M N|=|M||N|/|M ∩N|for any two subgroups M and N of G, one can prove the following.
Lemma 4.6.1 IfM,N are two normalπ-subgroups ofG, so is their product
By Lemma 4.6.1, there is a unique largest normalπ-subgroup ofGwhich is denoted byOπ(G). This subgroup is also characteristic. Note that|Oπ(G)|
is a divisor of|G|π, not necessarily to be equal. Similarly, replacing π by π0, we may define O
π0(G) to be the largest
normalπ0-subgroup ofGwhich is also characteristic and|O
π0(G)|is a divisor
of |G|π0.
It is easy to prove the following.
Lemma 4.6.2 Oπ(G/Oπ(G)) = 1and Oπ0(G/Oπ0(G)) = 1.
Next, we define a subgroupOππ0(G) byOππ0(G)/Oπ(G) =Oπ0(G/Oπ(G)),
and define Oππ0π(G) by Oππ0π(G)/Oππ0(G) = Oπ(G/Oππ0(G)), and so on.
Thus we get a series
1≤Oπ(G)≤Oππ0(G)≤Oππ0π(G)≤ · · ·, (4.3)
called the π-series of G.
Note that Oππ0(G) is the inverse image of the subgroup Oπ0(G/Oπ(G))
under the canonical projectionG→G/Oπ(G),and henceOππ0(G) is a nor-
mal subgroup ofG. So the series (4.3) is a normal series.
This series plays an important role in the study of solvable groups. Definition 4.6.1 Let π be a set of primes.
(1) A finite group Gis called π-separableif there is a normal series ofG G=N0≥N1≥N2≥ · · · ≥Nr= 1, (4.4)
such thatNi/Ni+1 is aπ-group or aπ0-group, for i= 0,1, . . . , r−1. (2) The group G is called π-solvable if there is a normal series (4.4) of
G such that Ni/Ni+1 is a π0-group or a p-group for some p ∈ π, for i= 0,1, . . . , r−1.
Obviously, G being π-separable is equivalent to G being π0-separable. However,Gbeingπ-solvable is not equivalent toG beingπ0-solvable. (Give an example to show this.)
On the other hand, by Feit-Thompson Theorem, if 2 ∈/ π, then π- separable groups are also π-solvable. Moreover, for any prime p, G being p-separable is equivalent to Gbeingp-solvable.
Solvable groups 163
Proposition 4.6.3 A finite group G is π-separable if and only if the π- series ofG terminates in G.
The next two theorem are basic for π-separable groups and π-solvable groups.
Theorem 4.6.4 Let Gbe a π-separable group (or π-solvable group). Then (1) Each subgroup ofG is π-separable (or π-solvable);
(2) Each quotient group ofG is π-separable (or π-solvable);
(3) The minimal normal subgroups ofGareπ0-group orπ-group (π0-group
or p-group where p∈π);
(4) For any N CG, N 6= G, we have Oπ(G/N) 6= 1 or Oπ0(G/N) 6= 1
(Oπ0(G/N)6= 1 or Op(G/N)6= 1 for some p∈π.)
Proof: We prove this theorem only for π-separable groups. Since G is π-separable, G has a normal series (4.4) such that Ni/Ni+1 is π-group or π0-group for any i.
(1) AssumeH ≤G. Then
H =H∩N0 ≥H∩N1≥ · · · ≥H∩Nr= 1 is a normal series ofH and
H∩Ni/H∩Ni+1 = H∩Ni/(H∩Ni)∩Ni+1 ∼
= (H∩Ni)Ni+1/Ni+1 is a subgroup of Ni/Ni+1 and hence is a π-group or π0-group.
(2) AssumeHCG andH 6=G. Then
G/H=N0H/H ≥N1H/H ≥ · · · ≥NrH/H = 1
is a normal series ofG/H and
NiH/H /Ni+1H/H ∼= NiH/Ni+1H ∼
= Ni/Ni∩Ni+1H
is a quotient group of Ni/Ni+1, and hence isπ-group or π0-group.
(3) By definition ofπ-separable groups, each chief factor of Gis either a π-group or π0-group, so is each minimal normal subgroup of G.
(4) Since G/N is π-separable, every minimal normal subgroup M/N of G/N is either π-group or π0-group. Hence either O
π(G/N) 6= 1, or
Oπ0(G/N)6= 1. ¤
The next theorem gives some sufficient conditions for a group to be π- separable orπ-solvable.
Theorem 4.6.5 A group G is π-separable (orπ-solvable) if one of the fol- lowing holds:
(1) There is a π-separable (or π-solvable) normal subgroup N of G such thatG/N is π-separable (or π-solvable);
(2) For any NCG with N 6=G, each minimal normal subgroup of G/N is either aπ-group (or a p-group for p∈π), or a π0-group;
(3) For any NCG with N 6=G, either Oπ(G/N) 6= 1 (or Op(G/N) 6= 1
for p∈π) or Oπ0(G)6= 1.
Proof: Again, we prove this theorem only forπ-separable groups.
(1) Since G/N isπ-separable, there is a normal series fromGtoN such that every factor group is either a π-group or a π0-group. Since N is π- separable, theπ-series ofN terminates toN. (Note that in this series every term is a characteristic subgroup of N). Connect these two series. The resulting series is a normal series of G whose factor groups are π-group or π0-group, hence Gis π-separable.
(2) Use induction on |G|. Let K be a minimal normal subgroup of G. Then by the assumption, K is a π-group or a π0-group. Consider G/K. Obviously, the conditions of the theorem holds for G/K and |G/K|<|G|, by the induction hypothesis, G/K is π-separable.
(3) Use induction on|G|. By assumption, eitherOπ(G)6= 1 orOπ0(G)6=
1. Without loss of generality we may assume that Oπ(G) 6= 1. Then
G/Oπ(G) also satisfies the conditions of the theorem. Since |G/Oπ(G)| <
|G|, by the induction hypothesis, G/Oπ(G) is π-separable. (1) gives the
π-separability ofG. ¤
The next theorem gives some characterizations for solvable groups by using π-separable andπ-solvable groups.
Solvable groups 165
(1) G is solvable;
(2) For any setπ of primes, G isπ-separable (or π-solvable); (3) For any prime numberp, G is p-solvable;
(4) For any N EG with G/N 6= 1, there is a prime number p such that Op(G/N)6= 1;
(5) For any N EG with G/N 6= 1, there is a characteristic subgroup K/N 6= 1 of G/N such that K/N is an elementary abelian p-group.
Proof: (1)⇒(2): By Theorem 3.1.4, any chief factor of a solvable group Gis an elementary abelian p-group. So, by Definition 4.6.1,Gis π-solvable (and alsoπ-separable) for any π.
(2)⇒(3): Obvious.
(3)⇒(4): Assume thatM/Nis a minimal normal subgroup ofG/N such that the setπ of prime divisors of|M/N|has minimum possible cardinality. Since G is assumed to be p-solvable for any p,M/N is also p-solvable. We claim that π = {p} is a one-element set. If there is a prime q 6= p with q ∈ π, then by the p-solvability of M/N, we have either Op(M/N) 6= 1 or
Op0(M/N)6= 1, however both are proper subgroups ofM/N. Since these two
subgroups are characteristic subgroups of M/N, they are normal in G/N. This contradicts the minimality ofM/N.
(4)⇒(5): By (4) one may assume that for some primep,Op(G/N)6= 1.
Then, the center Z := Z(Op(G/N)) is a nontrivial characteristic subgroup
of thep-group Op(G/N) and so it is characteristic inG/N. Its Frattini sub-
group is Φ(Z) = Z0f
1(Z) = f1(Z). As the first case, assume Φ(Z) = 1. Let K be the inverse image of the center Z under the canonical projection G→G/N. ThenK/N ∼=Z=Z/Φ(Z) is elementary abelian and character- istic inG/N.Next, let Φ(Z) =f1(Z)6= 1.Since thisp-group has an element of orderp, we have Φ(f1(Z)) =f1(f1(Z)) f1(Z).If Φ(f1(Z)) = 1, then one can takeKas the inverse image off1(Z) under the canonical projection G→ G/N as the previous case. If Φ(f1(Z))6= 1, repeat the same process withZ1 =f1(Z). Finally, one can get a desired characteristic subgroup.
(5) ⇒ (1): By (5), G has a normal series such that each factor is an elementary abelian p-group, and henceG is solvable. ¤
Recall that a Hall π-subgroup of G is a π-subgroup whose order and index are coprime.
The next theorem can be viewed as a generalization of the Sylow theo- rems for solvable groups.
Theorem 4.6.7 (P. Hall) Let G be a π-separable group. Then (1) there exists at least one Hall π-subgroup of G.
(2) Any two Hall π-subgroups are conjugate inG.
(3) Any π-subgroup of Gis contained in a Hall π-subgroup.
Proof: (1) SinceGisπ-separable, either Oπ(G)6= 1 orOπ0(G)6= 1. With-
out loss of generality one may assumeOπ(G)6= 1. LetG=G/Oπ(G). By the
induction, one can assume thatGhas a Hallπ-subgroupH/Oπ(G) and a Hall
π0-subgroup U/O
π(G). Then |H/Oπ(G)|= |G/Oπ(G)|π and |U/Oπ(G)| =
|G/Oπ(G)|π0. From the first equality, we have|H|=|Oπ(G)||G/Oπ(G)|π =
|G|π, that is, H is a Hall π-subgroup of G. From the second equality, one
can see that |U|π0 = |G|π0 and Oπ(G) is a normal Hall π-subgroup of U.
By Schur-Zassenhaus Theorem, Oπ(G) has a complement, say K, inU and
then the complement K is a Hall π0-subgroup of U, which is also a Hall π0-subgroup ofG.
(2) Let H1 and H2 be any two Hall π-subgroups of G. First assume that Oπ(G) 6= 1. Then Hi/Oπ(G) is a Hall π-subgroups of G/Oπ(G) for i= 1,2 and then by inductionH1 =H2g for some g∈G. Next, assume that Oπ0(G) 6= 1. ThenHiOπ0(G)/Oπ0(G) is a Hall π-subgroup of G/Oπ0(G) for
i= 1,2 and then by inductionH1Oπ0(G) =H2hOπ0(G) for someh∈G.Since
Oπ0(G) is a normal Hall π0-subgroup of H1Oπ0(G), by Schur-Zassenhuas
Theorem, there is a k ∈Oπ0(G) such that H1 = (H2h)k =H2hk, the desired
result.
(3) Use induction on |G|. Let K be a π-subgroup of G. Since G is π-separable, either (i)Oπ(G)6= 1 or (ii)Oπ0(G)6= 1.
(i) Oπ(G) 6= 1: LetG=G/Oπ(G). By the induction hypothesis, the π-
subgroupKOπ(G)/Oπ(G) ofGis contained in a Hallπ-subgroupH/Oπ(G)
of G. Hence KOπ(G) ≤ H and K ≤ H. Note that H is also a Hall π-
subgroup of G, (3) holds.
(ii) Oπ0(G) 6= 1: Let G = G/Oπ0(G). By the induction hypothesis,
the π-subgroup KOπ0(G)/Oπ0(G) of G is contained in a Hall π-subgroup
M/Oπ0(G) of G. Hence KOπ0(G) ≤ M. If M < G, by the induction
hypothesis, K is contained in a Hall π-subgroup H of M. It is easy to check that the Hall π-subgroups of M are also Hall π-subgroups of G. If M = G, take a Hall π-subgroup H of G. Let K1 = KOπ0(G)∩H. Since
G=Oπ0(G)·H =KOπ0(G)·H, we have |G|= |K||Oπ0(G)||H| |KOπ0(G)∩H| = |K||G| |K1| ,
Solvable groups 167
and hence|K1|=|K|. This shows thatK andK1 are two Hallπ-subgroups of G. By Schur-Zassenhaus Theorem, there is an x ∈ KOπ0(G) such that
Kx =K
1=KOπ0(G)∩H, and hence Kx≤H, K ≤Hx−1. ¤
Note that if π ={p} a prime, Theorem 4.6.7 is nothing but the Sylow Theorem. The following result due to P. Hall gives a characterization of finite solvable groups.
Theorem 4.6.8 (P. Hall)
(1) Let G be solvable. Then for any set π of primes, G has a Hall π- subgroup, and any two Hall π-subgroups are conjugate. Also, any π- subgroup is contained in a Hall π-subgroup of G.
(2) Let |G|=pα1
1 · · ·pαss. Then G is solvable if and only if G has a Hall
p0i-subgroup for every i= 1, . . . , s.
Proof: (1) By Theorems 4.6.6 and 4.6.7. (2) We only need to prove the “only if” part.
Ifs= 1, that isGis ap-group, thenGis solvable. Ifs= 2, Theorem 2.3.5 in the next chapter gives the solvability ofG. So we may assume thats≥3. Use induction on |G|, we divide the proof into two steps:
Step (1). For i= 1, . . . , s, letHi be a Hallp0
i-subgroup ofG. ThenHi is
solvable: Since |G:Hi|=pαii, fori6=j we have (|G:Hi|,|G:Hj|) = 1. By
Proposition 1.1.6(3),|G:Hi∩Hj|=pαiipjαj, and hence|Hi:Hi∩Hj|=pαjj.
This shows that Hi ∩Hj is a Hall p0
j-subgroup of Hi. By the induction
hypothesis,Hi is solvable.
Step (2). Proof of the solvability ofG: Consider three solvable subgroups H1, H2, H3 of G. There is a prime number p such that M =Op(H1) 6= 1. Since (|G : H2|,|G : H3|) = 1, p divides at least one of |H2| and |H3|. Without loss of generality we may assume that p | |H2| and P ≤ H2 is a Sylowp-subgroup of G. By Sylow Theorems, there exists ag∈Gsuch that M ≤Pg ≤Hg
2. By Proposition 1.1.6(3) again, G=H1H2g. For any element x ∈G, one may let x =x1x2 where x1 ∈H1 and x2 ∈ H2g. Since M CH1 and M ≤H2g,
Mx =Mx1x2 =Mx2 ≤Hg 2. LettingN =hMx|x∈Gi, we haveN ≤Hg
2. Thus 16=NCG. SinceH2g is solvable, so is N. Moreover, HiN/N is a Hall p0
i-subgroup of G/N, by the
Corollary 4.6.9 A finite group Gis solvable if and only if whenever |G|= mn with (m, n) = 1, Ghas a subgroup order m.
Exercises
4.6.1. A finite solvable group has a maximal subgroup which is normal. 4.6.2. Let M be a maximal subgroup of a π-separable group G. Then the
prime factors of|G:M|are all inπ, or all in π0.
4.6.3. Let G be a π-separable group. If Oπ0(G) = 1, then CG(Oπ(G)) ≤
Oπ(G).
4.6.4. LetGbe a group of order 60. If its Sylow 5-subgroups are not normal, thenG∼=A5.
4.6.5. (Lagrange groups) A finite groupG is called a Lagrange group if, for any divisordof |G|,Ghas at least one subgroup of orderd.
(1) Finitep-groups are Lagrange groups. (Prove it.)
(2) Finite abelian and nilpotent groups are Lagrange groups.
(3) Lagrange groups are solvable. (For a proof, see Theorem 4.6.8(2).) But the converse is not true.
(4) The property of a group being Lagrange is inherited by neither subgroups nor quotient groups. (Give examples.)
(5) Any solvable group can be embedded in a Lagrange group.
Moreover, any solvable group can be embedded in a Lagrange group which is direct-indecomposable, i.e., which cannot be expressed into a direct product of two nontrivial subgroups. See Gagen (1977).