Nil Algebras and Periodic Groups

Counting the Periodic Groups Generated by Two Finite Groups

“…We recall the Golod-Schafarevitch construction of infinite finitely generated p-groups, ( [3] or [1]). For any integer d ^ 2 and any prime p let R(d, p) be the ring of formal power series in the non-commuting indeterminates x l ,...,x d with coefficients in GF{p).…”

Turing Machines on Graphs and Inescapable Groups

“…Golod used this fact to establish the existence of a Burnside group, that is, an infinite group in which every element has finite order. He did more than this, in fact, and produced an infinite p-group for each prime, p. The diagonalization in Section 4.1.4 will follow the same general ideas as Golod's construction as simplified for countable fields in Fischer and Struik [4]. 4.1.4.…”

“…The Construction. In Fischer and Struik [4], there is a construction of a nil-algebra over finite and countable fields. Although not expressly stated there, the construction is essentially a diagonalization over all polynomials.…”

On Periodic Generalized Nilpotent Groups

“…An easy computation shows that this condition is satisfied, with e = £, if r, = 0 for 1 ^ 10 and r,-^ 2 for i ^ 11. We refer the reader to Golod [4] or Fischer and Struik [3] for the (by now well known) arguments which will show that there is a set P x of homogeneous elements, containing no elements of degree less than 11 and at most one of each greater degree such that, in the ring A = R/RP^R, the ideal B generated by the images x, y of x and y is a nil ring.…”