Golod–Shafarevich theorem

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In mathematics, the Golod–Shafarevich theorem was proved in 1964 by two Russian mathematicians, Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which has consequences in various branches of algebra.

The inequality

Let A = K<x1, ..., xn> be the free algebra over a field K in n = d + 1 non-commuting variables xi.

Let J be the 2-sided ideal of A generated by homogeneous elements fj of A of degree dj with

2 ≤ d1d2 ≤ ...

where dj tends to infinity. Let ri be the number of dj equal to i.

Let B=A/J, a graded algebra. Let bj = dim Bj.

The fundamental inequality of Golod and Shafarevich states that

As a consequence:

  • B is infinite-dimensional if rid2/4 for all i
  • if B is finite-dimensional, then ri > d2/4 for some i.

Applications

This result has important applications in combinatorial group theory:

  • If G is a nontrivial finite p-group, then r > d2/4 where d = dim H1(G,Z/pZ) and r = dim H2(G,Z/pZ) (the mod p cohomology groups of G). In particular if G is a finite p-group with minimal number of generators d and has r relators in a given presentation, then r > d2/4.
  • For each prime p, there is an infinite group G generated by three elements in which each element has order a power of p. The group G provides a counterexample to the generalised Burnside conjecture: it is a finitely generated infinite torsion group, although there is no uniform bound on the order of its elements.

In class field theory, the class field tower of a number field K is created by iterating the Hilbert class field construction. Another consequence of the construction is that such towers may be infinite (in other words, do not always terminate in a field equal to its Hilbert class field). Specifically,

  • Let K be an imaginary quadratic field whose discriminant has at least 6 prime factors. Then the maximal unramified 2-extension of K has infinite degree.

More generally, a number field with sufficiently many prime factors in the discriminant has an infinite class field tower.

References

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  • Johnson, D.L. (1980). "Topics in the Theory of Group Presentations" (1st ed.). Cambridge University Press. ISBN 0-521-23108-6. See chapter VI.
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  • Serre, J.-P. (2002), "Galois Cohomology," Springer-Verlag. ISBN 3-540-42192-0. See Appendix 2. (Translation of Cohomologie Galoisienne, Lecture Notes in Mathematics 5, 1973.)