with group action , and minimal polynomial containing the 5th roots of unity excluding .
with group action , and minimal polynomial .
, where is the identity permutation. All of the defining group actions change a single extension while keeping all of the other extensions fixed. For example, an element of this group is the group action . A general element in the group can be written as for a total of 80 elements.
It is worthwhile to note that this group is not abelian itself. For example:
In fact, in this group, . The solvable group is isometric to , defined using the semidirect product and direct product of the cyclic groups. In the solvable group, is not a normal subgroup.
meaning that Gj−1isnormalinGj, such that Gj/Gj−1 is an abelian group, for j = 1, 2, ..., k.
Or equivalently, if its derived series, the descending normal series
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 subgroupNofH, the quotient H/N is abelian if and only ifN includes the commutator subgroup of H. The least n such that G(n) = 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 groupsofprimeorder. 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 Zofintegers under addition is isomorphictoZ 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.
A small example of a solvable, non-nilpotent group is the symmetric groupS3. 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 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.
for some field . Then, the group quotient can be found by taking arbitrary elements in , multiplying them together, and figuring out what structure this gives. So
Note the determinant condition on implies , hence is a subgroup (which are the matrices where ). For fixed , the linear equation implies , which is an arbitrary element in since . Since we can take any matrix in and multiply it by the matrix
with , we can get a diagonal matrix in . This shows the quotient group .
For a linear algebraic group, a Borel subgroup is defined as a subgroup which is closed, connected, and solvable in , and is a maximal possible subgroup with these properties (note the first two are topological properties). For example, in and the groups of upper-triangular, or lower-triangular matrices are two of the Borel subgroups. The example given above, the subgroup in, is a Borel subgroup.
Solvability is closed under a number of operations.
IfG is solvable, and H is a subgroup of G, then H is solvable.[2]
IfG is solvable, and there is a homomorphism from GontoH, 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.[3]
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 productG × H is solvable.
IfH 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:
IfG and H are solvable, and X is a G-set, then the wreath productofG 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.
As a strengthening of solvability, a group G is called supersolvable (orsupersoluble) 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:
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 (Malcev 1949).
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.
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