In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.
Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equations. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable[1] (note this theorem holds only in characteristic 0). This means associated to a polynomial
f\inF[x]
such thatF=F0\subseteqF1\subseteqF2\subseteq … \subseteqFm=K
Fi=Fi-1[\alphai]
mi | |
\alpha | |
i |
\inFi-1
\alphai
mi | |
x |
-a
a\inFi-1
Fm
f(x)
For example, the smallest Galois field extension of
Q
gives a solvable group. It has associated field extensionsa=\sqrt[5]{\sqrt{2}+\sqrt{3}}
giving a solvable group of Galois extensions containing the following composition factors:Q\subseteqQ(\sqrt{2})\subseteqQ(\sqrt{2},\sqrt{3})\subseteqQ(\sqrt{2},\sqrt{3})\left(e2i\pi/\right) \subseteqQ(\sqrt{2},\sqrt{3})\left(e2i\pi/,a\right)
Aut\left(Q(\sqrt{2)}\right/Q)\congZ/2
f\left(\pm\sqrt{2}\right)=\mp\sqrt{2}, f2=1
x2-2
Aut\left(Q(\sqrt{2,\sqrt{3})}\right/Q(\sqrt{2)})\congZ/2
g\left(\pm\sqrt{3}\right)=\mp\sqrt{3}, g2=1
x2-3
Aut\left(Q(\sqrt{2},\sqrt{3})\left(e2i\pi/\right)/ Q(\sqrt{2},\sqrt{3}) \right) \congZ/4
hn\left(e2im\pi/5\right)=e2(n+1)mi\pi/5, 0\leqn\leq3, h4=1
x4+x3+x2+x+1=(x5-1)/(x-1)
1
Aut\left(Q(\sqrt{2},\sqrt{3})\left(e2i\pi/,a\right)/ Q(\sqrt{2},\sqrt{3})\left(e2i\pi/\right) \right) \congZ/5
jl(a)=e2li\pi/5a, j5=1
x5-\left(\sqrt{2}+\sqrt{3}\right)
, where
1
fgh3j4
fagbhnjl, 0\leqa,b\leq1, 0\leqn\leq3, 0\leql\leq4
It is worthwhile to note that this group is not abelian itself. For example:
hj(a)=h(e2i\pi/5a)=e4i\pi/5a
jh(a)=j(a)=e2i\pi/5a
In fact, in this group,
jh=hj3
(C5\rtimes\varphiC4) x (C2 x C2), where \varphih(j)=hjh-1=j2
C4
A group G is called solvable if it has a subnormal series whose factor groups (quotient groups) are all abelian, that is, if there are subgroups
1=G0\triangleleftG1\triangleleft … \triangleleftGk=G
Or equivalently, if its derived series, the descending normal series
G\trianglerightG(1)\trianglerightG(2)\triangleright … ,
For finite groups, an equivalent definition is that a solvable group is a group with a composition series all of whose factors are cyclic groups of prime order. This is equivalent because a finite group has finite composition length, and every simple abelian group is cyclic of prime order. The Jordan–Hölder theorem guarantees that if one composition series has this property, then all composition series will have this property as well. For the Galois group of a polynomial, these cyclic groups correspond to nth roots (radicals) over some field. The equivalence does not necessarily hold for infinite groups: for example, since every nontrivial subgroup of the group Z of integers under addition is isomorphic to Z itself, it has no composition series, but the normal series, with its only factor group isomorphic to Z, proves that it is in fact solvable.
The basic example of solvable groups are abelian groups. They are trivially solvable since a subnormal series is formed by just the group itself and the trivial group. But non-abelian groups may or may not be solvable.
More generally, all nilpotent groups are solvable. In particular, finite p-groups are solvable, as all finite p-groups are nilpotent.
In particular, the quaternion group is a solvable group given by the group extension
where the kernel1\toZ/2\toQ\toZ/2 x Z/2\to1
Z/2
-1
Group extensions form the prototypical examples of solvable groups. That is, if
G
G'
defines a solvable group1\toG\toG''\toG'\to1
G''
A small example of a solvable, non-nilpotent group is the symmetric group S3. In fact, as the smallest simple non-abelian group is A5, (the alternating group of degree 5) it follows that every group with order less than 60 is solvable.
The Feit–Thompson theorem states that every finite group of odd order is solvable. In particular this implies that if a finite group is simple, it is either a prime cyclic or of even order.
The group S5 is not solvable - it has a composition series (and the Jordan–Hölder theorem states that every other composition series is equivalent to that one), giving factor groups isomorphic to A5 and C2; and A5 is not abelian. Generalizing this argument, coupled with the fact that An is a normal, maximal, non-abelian simple subgroup of Sn for n > 4, we see that Sn is not solvable for n > 4. This is a key step in the proof that for every n > 4 there are polynomials of degree n which are not solvable by radicals (Abel–Ruffini theorem). This property is also used in complexity theory in the proof of Barrington's theorem.
Consider the subgroups
for some fieldofB=\left\{\begin{bmatrix} *&*\\ 0&* \end{bmatrix}\right\}, U=\left\{\begin{bmatrix} 1&*\\ 0&1 \end{bmatrix}\right\}
GL2(F)
F
B/U
B,U
Note the determinant condition on\begin{bmatrix} a&b\\ 0&c\end{bmatrix} ⋅ \begin{bmatrix} 1&d\\ 0&1\end{bmatrix}= \begin{bmatrix} a&ad+b\\ 0&c \end{bmatrix}
GL2
ac ≠ 0
F x x F x \subsetB
b=0
a,b
ad+b=0
d=-b/a
F
b\inF
B
with\begin{bmatrix} 1&d\\ 0&1 \end{bmatrix}
d=-b/a
B
B/U\congF x x F x
Notice that this description gives the decomposition of
B
F\rtimes(F x x F x )
(a,c)
b
(a,c)(b)=ab
(a,c)(b+b')=(a,c)(b)+(a,c)(b')=ab+ab'
corresponds to the element\begin{bmatrix} a&b\\ 0&c \end{bmatrix}
(b) x (a,c)
G
G
GLn
SLn
B
GL2
In
GL3
NoticeB=\left\{ \begin{bmatrix} *&*&*\\ 0&*&*\\ 0&0&* \end{bmatrix} \right\}, U1=\left\{ \begin{bmatrix} 1&*&*\\ 0&1&*\\ 0&0&1 \end{bmatrix} \right\}
B/U1\congF x x F x x F x
U\rtimes (F x x F x x F x )
In the product group
GLn x GLm
where\begin{bmatrix} T&0\\ 0&S \end{bmatrix}
T
n x n
S
m x m
Any finite group whose p-Sylow subgroups are cyclic is a semidirect product of two cyclic groups, in particular solvable. Such groups are called Z-groups.
Numbers of solvable groups with order n are (start with n = 0)
0, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 12, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, ...
Orders of non-solvable groups are
60, 120, 168, 180, 240, 300, 336, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 900, 960, 1008, 1020, 1080, 1092, 1140, 1176, 1200, 1260, 1320, 1344, 1380, 1440, 1500, ...
Solvability is closed under a number of operations.
Solvability is closed under group extension:
It is also closed under wreath product:
For any positive integer N, the solvable groups of derived length at most N form a subvariety of the variety of groups, as they are closed under the taking of homomorphic images, subalgebras, and (direct) products. The direct product of a sequence of solvable groups with unbounded derived length is not solvable, so the class of all solvable groups is not a variety.
See main article: Burnside's theorem. Burnside's theorem states that if G is a finite group of order paqb where p and q are prime numbers, and a and b are non-negative integers, then G is solvable.
See main article: supersolvable group. As a strengthening of solvability, a group G is called supersolvable (or supersoluble) if it has an invariant normal series whose factors are all cyclic. Since a normal series has finite length by definition, uncountable groups are not supersolvable. In fact, all supersolvable groups are finitely generated, and an abelian group is supersolvable if and only if it is finitely generated. The alternating group A4 is an example of a finite solvable group that is not supersolvable.
If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups:
cyclic < abelian < nilpotent < supersolvable < polycyclic < solvable < finitely generated group.
A group G is called virtually solvable if it has a solvable subgroup of finite index. This is similar to virtually abelian. Clearly all solvable groups are virtually solvable, since one can just choose the group itself, which has index 1.
A solvable group is one whose derived series reaches the trivial subgroup at a finite stage. For an infinite group, the finite derived series may not stabilize, but the transfinite derived series always stabilizes. A group whose transfinite derived series reaches the trivial group is called a hypoabelian group, and every solvable group is a hypoabelian group. The first ordinal α such that G(α) = G(α+1) is called the (transfinite) derived length of the group G, and it has been shown that every ordinal is the derived length of some group .
A finite group is p-solvable for some prime p if every factor in the composition series is a p-group or has order prime to p. A finite group is solvable iff it is p-solvable for every p. [4]