Solvable group

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 (note this theorem holds only in characteristic 0). This means associated to a polynomial \(f \in F[x]\) there is a tower of field extensions

such that

1. \(F_i = F_{i-1}[\alpha_i]\) where \(\alpha_i^{m_i} \in F_{i-1}\), so \(\alpha_i\) is a solution to the equation \(x^{m_i} - a\) where \(a \in F_{i-1}\)
2. \(F_m\) contains a splitting field for \(f(x)\)

The smallest Galois field extension of \(\mathbb{Q}\) containing the element

gives a solvable group. The associated field extensions

give a solvable group of Galois extensions containing the following composition factors (where \(1\) is the identity permutation).

• \(\mathrm{Aut}\left(\mathbb{Q(\sqrt{2})}\right/\mathbb{Q}) \cong \mathbb{Z}/2\) with group action \(f\left(\pm\sqrt{2}\right) = \mp\sqrt{2}, \ f^2 = 1\), and minimal polynomial \(x^2 - 2\)
• \(\mathrm{Aut}\left(\mathbb{Q(\sqrt{2},\sqrt{3})}\right/\mathbb{Q(\sqrt{2})}) \cong \mathbb{Z}/2\) with group action \(g\left(\pm\sqrt{3}\right) = \mp\sqrt{3} ,\ g^2 = 1\), and minimal polynomial \(x^2 - 3\)
• \(\mathrm{Aut}\left( \mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}\right)/ \mathbb{Q}(\sqrt{2}, \sqrt{3}) \right) \cong \mathbb{Z}/4\) with group action \(h^n\left(e^{2im\pi/5}\right) = e^{2^{n}im\pi/5} , \ 0 \leq n \leq 3, \ h^4 = 1\), and minimal polynomial \(x^4 + x^3+x^2+x+1 = (x^5 - 1)/(x-1)\) containing the 5th roots of unity excluding \(1\)
• \(\mathrm{Aut}\left( \mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}, a\right)/ \mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}\right) \right) \cong \mathbb{Z}/5\) with group action \(j^l(a) = e^{2li\pi/5}a, \ j^5 = 1\), and minimal polynomial \(x^5 - \left(\sqrt{2} + \sqrt{3}\right)\)

Each of the defining group actions (for example, \(fgh^3j^4\)) changes a single extension while keeping all of the other extensions fixed. The 80 group actions are the set \(\{f^ag^bh^nj^l,\ 0 \leq a, b \leq 1,\ 0 \leq n \leq 3,\ 0 \leq l \leq 4 \}\).

This group is not abelian. For example, \(hj(a) = h(e^{2i\pi/5}a) = e^{4i\pi/5}a\), whilst \(jh(a) = j(a) = e^{2i\pi/5}a\), and in fact, \(jh = hj^3\).

It is isomorphic to \((\mathbb{Z}_5 \rtimes_\varphi \mathbb{Z}_4) \times (\mathbb{Z}_2 \times \mathbb{Z}_2)\), where \(\varphi_h(j) = hjh^{-1} = j^2\), defined using the semidirect product and direct product of the cyclic groups. \(\mathbb{Z}_4\) is not a normal subgroup.

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

\(1 = G_0\triangleleft G_1 \triangleleft \cdots \triangleleft G_k=G\)

meaning that G_j−1 is normal in G_j, such that G_j /G_j−1 is an abelian group, for j = 1, 2, ..., k.

Or equivalently, if its derived series, the descending normal series

\(G\triangleright G^{(1)}\triangleright G^{(2)} \triangleright \cdots,\)

where every subgroup is the commutator subgroup of the previous one, eventually reaches the trivial subgroup of G. These two definitions are equivalent, since for every group H and every normal subgroup N of H, the quotient H/N is abelian if and only if N includes the commutator subgroup of H. The least n such that G = 1 is called the derived length of the solvable group G.

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 {0, Z}, 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.

Group extensions form the prototypical examples of solvable groups. That is, if \(G\) and \(G'\) are solvable groups, then any extension

defines a solvable group \(G''\). In fact, all solvable groups can be formed from such group extensions.

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 {E, A₅, S₅} (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 A_n is a normal, maximal, non-abelian simple subgroup of S_n for n > 4, we see that S_n 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 field \(\mathbb{F}\). Then, the group quotient \(B/U\) can be found by taking arbitrary elements in \(B,U\), multiplying them together, and figuring out what structure this gives. So

Note the determinant condition on \(GL_2\) implies \(ac \neq 0\), hence \(\mathbb{F}^\times \times \mathbb{F}^\times \subset B\) is a subgroup (which are the matrices where \(b=0\)). For fixed \(a,b\), the linear equation \(ad + b = 0\) implies \(d = -b/a\), which is an arbitrary element in \(\mathbb{F}\) since \(b \in \mathbb{F}\). Since we can take any matrix in \(B\) and multiply it by the matrix

with \(d = -b/a\), we can get a diagonal matrix in \(B\). This shows the quotient group \(B/U \cong \mathbb{F}^\times \times \mathbb{F}^\times\).

For a linear algebraic group \(G\), a Borel subgroup is defined as a subgroup which is closed, connected, and solvable in \(G\), and is a maximal possible subgroup with these properties (note the first two are topological properties). For example, in \(GL_n\) and \(SL_n\) the groups of upper-triangular, or lower-triangular matrices are two of the Borel subgroups. The example given above, the subgroup \(B\) in \(GL_2\), is a Borel subgroup.

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, ... (sequence A201733 in the OEIS)

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, ... (sequence A056866 in the OEIS)

Solvability is closed under a number of operations.

• If G is solvable, and H is a subgroup of G, then H is solvable.
• If G is solvable, and there is a homomorphism from G onto H, then H is solvable; equivalently (by the first isomorphism theorem), if G is solvable, and N is a normal subgroup of G, then G/N is solvable.
• The previous properties can be expanded into the following "three for the price of two" property: G is solvable if and only if both N and G/N are solvable.
• In particular, if G and H are solvable, the direct product G × H is solvable.

Solvability is closed under group extension:

• If H and G/H are solvable, then so is G; in particular, if N and H are solvable, their semidirect product is also solvable.

It is also closed under wreath product:

• If G and H are solvable, and X is a G-set, then the wreath product of G and H with respect to X is also solvable.

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.