Edward Formanek

The automorphism group of a free group is not linear

A linear group has ascending chain condition on centralizers. (Malcev [S, p. 511). Finitely generated linear groups are residually finite. (Malcev [S, p. 453). A solvable linear group is nilpotent-by-(abelian-byfinite). (Tits [5, pp. 14551461). A linear group either contains a free group of rank two, or is solvable-by-(locally finite). Lubotzky [3] has characterized finitely generated linear groups over a field of characteristic zero by purely group theoretic conditions. These conditions appear difficult to check, although they have been used to show linearity for certain groups. However, we do not use his theorem. The main result of this paper (Theorem 5) is that for n 2 3, the automorphism group of a free group of rank n is not a linear group. The proof uses the representation theory of algebraic groups to show that a kind of “diophantine equation” between the irreducible representations of

Polynomial identity rings

A Combinatorial Aspects in PI-Rings.- Vesselin Drensky.- 1 Basic Properties of PI-algebras.- 2 Quantitative Approach to PI-algebras.- 3 The Amitsur-Levitzki Theorem.- 4 Central Polynomials for Matrices.- 5 Invariant Theory of Matrices.- 6 The Nagata-Higman Theorem.- 7 The Shirshov Theorem for Finitely Generated PI-algebras.- 8 Growth of Codimensions of PI-algebras.- B Polynomial Identity Rings.- Edward Formanek.- 1 Polynomial Identities.- 2 The Amitsur-Levitzki Theorem.- 3 Central Polynomials.- 4 Kaplanskys Theorem.- 5 Theorems of Amitsur and Levitzki on Radicals.- 6 Posners Theorem.- 7 Every PI-ring Satisfies a Power of the Standard Identity.- 8 Azumaya Algebras.- 9 Artins Theorem.- 10 Chain Conditions.- 11 Hilbert and Jacobson PI-Rings.- 12 The Ring of Generic Matrices.- 13 The Generic Division Ring of Two 2 x 2 Generic Matrices.- 14 The Center of the Generic Division Ring.- 15 Is the Center of the Generic Division Ring a Rational Function Field?.

A question of B. Plotkin about the semigroup of endomorphisms of a free group

Let F be a free group of finite rank n > 2, let End(F) be the semigroup of endomorphisms of F, and let Aut(F) be the group of automorphisms of F.

A conjecture of Regev about the Capelli polynomial

Abstract Let X1,…, Xn2, Y1,…, Yn2 be generic n × n matrices over a field k of characteristic zero. If ƒ(X 1,… , X n ) is a multilinear invariant of X1,…, Xn then ∑ πϱϵSπZ 2 (sign π ) f(X 1 Y π(1) ,X 2 Y π(2) Y π(3) Y π(4) , X 3 Y π(5) …Y π(9) …X n Y π(n 2 −2n+2) …Y π(n 2 ) )=f(X 1 …X n )Δ(Y) , where Sn2 is the symmetric group of degree n2, Δ(Y) is the discriminant of Y1,…, Yn2 and \ tf(X 1,… , X n ) is a uniquely defined multilinear invariant of X1,…, Xn. Thus ƒ → \ tf defines a function from the vector space of multilinear invariants of X1,…, Xn to itself. An analysis of this function is used to prove Regevs conjecture that ∑ πϱϵSπZ 2 ( sign πϱ)X n(1) Y ϱ(1) X π(2) X π(3) X π(4) Y ϱ(2) Y ϱ(3) Y ϱ(4) X π(5) … X π(n 2 −2π+2) …X π(n 2 ) Y ϱ(n 2 −2n+2) …Y ϱ(n 2 ) is nonzero. In addition, a variant of the above function is used to evaluate the Capelli polynomial.

Noetherian Pi-Rings

§1 Statement of Results Rings are associative and have a unit and subrings are assumed to have the same unit. Ring homomorphisms are unitary, as are modules. Ideal means two-sided ideal.

Hilbert series of fixed free algebras and noncommutative classical invariant theory

We compute an asymptotic formula for the number of invariants of a given degree, which compares nicely with the corresponding result in the classical commutative case. In Section 6 we show that if G is an infinite cyclic group generated by a unipotent matrix then

The ring of generic matrices

In Chapter 3, we defined a single n x n generic matrix. We now introduce the ring generated by a set of n x n generic matrices.

Characterizing a free group in its automorphism group

for beA and gEG. If the center of G is trivial, then i: G + A(G) is one-to-one, and G can be identified with a normal subgroup of A(G), with the action on G given by conjugation. A group G is said to be complete if i: G + A(G) is an isomorphism. A theorem of Burnside [Ru, p. 951 says that if G is a group with trivial center, then A(G) is complete if and only if G is a characteristic subgroup of A(G). Let F be a free group of finite rank n > 2. In [D-F] Joan Dyer and 1 used Burnside’s criterion to show that A(F) is a complete group. More precisely, WC showed that if q4: A(F) + A(F) is an automorphism of A(F), then d(P) = F.

The Nagata-Higman Theorem

The Nagata-Higman theorem for the nilpotency of nil algebras of bounded index was proved in 1953 by Nagata [Nal] over a field of characteristic 0 and then in 1956 by Higman [Hg] in the general setup. Much later it was discovered that this theorem was first established in 1943 by Dubnov and Ivanov [DI] but their paper was overlooked by the mathematical community. The theorem has many applications to the theory of PI-algebras as well as to invariant theory and structure theory of rings. Since we consider nil and nilpotent algebras which are nonunitary, in this chapter we shall work with the free nonunitary algebra K +(X).

Profinite completions and isomorphic finite quotients

Let G be a group and {G,} be the set of all normal subgroups of finite index in G. Then the set {G/G,, Gob} of finite quotients G/G, of G together with the canonical projections gob : G/G, ---) G/Gb whenever G, c Gb is an inverse system. The inverse limit_ I@ G/G, of this system is called the profinite completion of G and is denoted by G. The group d can also be described as the closure of the image of G under the diagonal mapping d : GA n(G/G,) where G/G, is given the discrete topology and n(G/G,) has the product topology. In this description the elements of 6 are the elements (g,) E n (G/G,) which satisfy @&go) =gb whenever G, c Gb. The object of this paper is to prove the following result: