Chapter VII. Solvable Groups
FC16 — Humphreys, Linear Algebraic Groups.
This chapter contains 9 definitions, 5 theorems, 11 propositions, 7 lemmas, 7 corollaries, 1 example, 1 construction and 1 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition: commutator subgroup (A,B)
For subgroups A,B, (A,B) is the subgroup generated by commutators (x,y)=xyx{-1}y{-1}.
Lemma: finite central quotient gives finite derived group
If [G:Z(G)] is finite then (G,G) is finite.
Lemma: finite commutator set for normal subgroups gives finite commutator subgroup
If A,B are normal and the set of pairwise commutators {(a,b)} is finite, then (A,B) is finite.
Proposition: closedness and connectedness of algebraic commutator groups
For closed A,B≤G, if A is connected then (A,B) is closed and connected; if A,B are normal then (A,B) is closed and normal. In particular (G,G) is closed.
Definition: derived series and solvable group
The derived series is G^{(0)}=G, G{(i+1)}=(G{(i)},G^{(i)}); G is solvable if this reaches e, with least index its derived length.
Lemma: closure properties of solvable groups
Subgroups and homomorphic images of solvable groups are solvable; solvable extensions are solvable; products of normal solvable subgroups are solvable.
Example/result: upper triangular groups are solvable
The upper triangular matrix group is solvable, with its derived behavior controlled by powers of the strictly upper-triangular ideal.
Definition: descending central series and nilpotent group
γ^0G=G and γ{i+1}G=(G,γiG); G is nilpotent when some γ^nG=e.
Lemma: elementary properties of nilpotent groups
Subgroups and quotients of nilpotent groups are nilpotent; if G/Z(G) is nilpotent then G is; nontrivial nilpotent groups have nontrivial center, and proper subgroups lie properly in their normalizers.
Proposition: center and normalizers in connected nilpotent algebraic groups
A connected positive-dimensional nilpotent algebraic group has positive-dimensional center; proper closed H has dim H<dim N_G(H), so codimension-one closed subgroups are normal.
Definition: unipotent algebraic subgroup
A subgroup is unipotent when every element is unipotent in the Jordan-decomposition sense.
Theorem: common fixed vector for a unipotent group
A unipotent subgroup of GL(V), V≠0 finite-dimensional, has a nonzero vector fixed by every element.
Corollary: unipotent groups are triangularizable and nilpotent
Every unipotent subgroup of GL_n is conjugate into U(n,K), hence is nilpotent.
Theorem: Lie–Kolchin theorem
A connected solvable subgroup of GL(V), V≠0, has a common eigenvector; equivalently it is conjugate into an upper triangular group.
Corollary/result: connected solvable groups stabilize a full flag
Iterating Lie–Kolchin yields a full G-stable flag in every finite-dimensional rational module of a connected solvable group.
Proposition: global and infinitesimal centralizers for semisimple normalizers
If closed H≤GL_n is normalized by a semisimple s, then Lie(C_H(s)) is the 1-eigenspace of Ad s on h, and the complementary eigenspaces give the tangent space of the commutator image at e.
Corollary: semisimple conjugacy orbit as homogeneous space
For semisimple s normalizing H, the conjugacy class map H→Cl_H(s) identifies with H/C_H(s).
Corollary: transverse centralizer–commutator differential decomposition
For the centralizer C and commutator image M′ near s, the product C×M′→H has an isomorphism on tangent spaces at the identity pair.
Proposition: semisimple conjugacy classes are closed
If H≤GL_n is normalized by a semisimple element s, then the H-conjugacy class of s is closed.
Theorem: semisimple action on a connected unipotent group
If connected unipotent U is normalized by semisimple s, the centralizer C_U(s) and commutator image M give a bijective product decomposition C×M→U; C and M are connected/closed and the commutator map restricts bijectively on M.
Proposition: centralizers of d-group actions infinitesimally
If closed H≤G is normalized by a d-group D, then Lie(C_H(D))=h^D.
Corollary: equivariant epimorphisms on d-group fixed identity components
A D-equivariant epimorphism H→H′ maps (H^D)° onto (H′^D)°.
Proposition: connected fixed subgroup in connected unipotent groups
If connected unipotent U is normalized by a d-group D, then C_U(D) is connected.
Construction/result: semisimple quotient exact sequence for a connected solvable group
For connected solvable G, Jordan decomposition yields the exact structural relation between its semisimple part, unipotent part, and a torus quotient used throughout §19.
Proposition: nilpotent connected solvable groups
A connected solvable G is nilpotent iff its semisimple elements form a subgroup; then that subgroup is closed and G decomposes as its semisimple and unipotent factors.
Theorem: structure of connected solvable algebraic groups
For connected solvable G, G_u is closed connected normal and contains (G,G), with a normal codimension-one filtration; maximal tori are conjugate under G_u and multiplication T⋉G_u→G is an isomorphism of varieties/groups in the stated form.
Corollary: semisimple and unipotent elements lie in maximal subgroups
Every semisimple element of a connected solvable group lies in a maximal torus; every unipotent element lies in a maximal connected unipotent subgroup.
Proposition: semisimple subgroups of a connected solvable group
A subgroup consisting entirely of semisimple elements lies in a torus, and its centralizer and normalizer in G coincide and are connected.
Definition: radical and unipotent radical
R(G) is the largest connected normal solvable subgroup; R_u(G) is the largest connected normal unipotent subgroup.
Definition: semisimple and reductive groups via radicals
A connected nontrivial G is semisimple when R(G)=e and reductive when R_u(G)=e.
Lemma: radical of a reductive group
For reductive G, R(G)=Z(G)° is a torus and has finite intersection with (G,G).
Result: connected one-dimensional groups are commutative
Every connected one-dimensional algebraic group is commutative.
Definition: vector group
A vector group is an algebraic group isomorphic to G_a^n.
Definition: e-group
In positive characteristic, an e-group is a commutative unipotent algebraic group of exponent p.
Proposition: embedding e-groups into vector groups
Every e-group embeds as a closed subgroup of a vector group of dimension one larger.
Definition: p-polynomial
In characteristic p, a p-polynomial is a K-linear combination of p^r-th powers of coordinate variables.
Lemma: additive polynomials are p-polynomials
Every polynomial map K^n→K additive under vector addition is a p-polynomial.
Lemma: squarefree subgroup zero sets force p-polynomials
If a squarefree polynomial has zero set an additive subgroup of K^n, then it is a p-polynomial.
Proposition: codimension-one vector subgroups are p-polynomial kernels
Every closed codimension-one subgroup of a vector group is the zero set/kernel of a p-polynomial.
Proposition: coordinate reduction of a p-polynomial homomorphism
For n>1, a p-polynomial homomorphism K^n→G_a can be put by a vector-group automorphism into coordinates whose kernel contains the first coordinate line.
Corollary: normal form for connected vector subgroups
A closed connected subgroup of a vector group can be carried by an automorphism to a coordinate vector subgroup.
Theorem: classification of connected one-dimensional algebraic groups
Over an algebraically closed field of arbitrary characteristic, every connected one-dimensional algebraic group is isomorphic to G_a or G_m.