[Math] Problems from the Kourovka Notebook that undergraduate students can fully appreciate

abstract-algebrabig-listgroup-theorysoft-question

The Kourovka Notebook is a collection of open problems in Group
Theory.

My question is: could you point out some (a "big-list" of) problems [by referencing them] presented in this book that are, in principle, accessible to undegraduate students: i.e., problems that refer to (and possibly might be solved by applying) definitions, concepts, and theorems that are presented in a book like Herstein's Topics in Algebra (and then, by extension, in an abstract algebra course for undergraduates).

The aim of this question is to allow undergraduate students to have a better understanding of current research in algebra by letting them see concretely open problems that can be easily related to known concepts.

Best Answer

Problem 8.10(a) from the 8th edition (1982):

Is the group $G = \langle a, b \mid a^n=1, ab = b^3 a^3 \rangle$ finite or infinite for $n = 7$? All other cases known. See Archive, 7.7 and 8.10 b. (D. L. Johnson)

Remark:

  • for $n=3$ the group has the order 6 (should be an easy exercise for a student to check this by hand and show that it's cyclic)

  • for $n=6$ it has the order 9072 (perhaps not so easy to check this by hand, but can be done using computer).

  • for $n=7$, the computer calculation runs too long without an answer.

  • It is known that $G$ is infinite for:

    • $n = 15$ in [D. J. Seal, Proc. Roy. Soc. Edinburgh (A), 92 (1982), 181–192]
    • $n = 9$ (and $15$) in [M. I. Prishchepov, Commun. Algebra, 23 (1995), 5095–5117].

An example in GAP illustrates the problem:

gap> F:=FreeGroup("a","b");
<free group on the generators [ a, b ]>
gap> G:=F/ParseRelators(GeneratorsOfGroup(F),"a^3=1,ab=b^3*a^3");
<fp group on the generators [ a, b ]>
gap> Size(G); # could be easily done by hand
6
gap> G:=F/ParseRelators(GeneratorsOfGroup(F),"a^6=1,ab=b^3*a^3");
<fp group on the generators [ a, b ]>
gap> Size(G);
9072
gap> G:=F/ParseRelators(GeneratorsOfGroup(F),"a^7=1,ab=b^3*a^3");
<fp group on the generators [ a, b ]>
gap> IsFinite(G);
#I  Coset table calculation failed -- trying with bigger table limit
#I  Coset table calculation failed -- trying with bigger table limit
... GAP was interrupted ...

The message about the coset table calculation hitting the limit is often a slight hint towards the fact that it may be infinite, but that's far from being the evidence - it is still possible that the calculation will succeed after increasing the limit several times.

Thus, the problem for $n=7$ is still open...

Update: the answer to this question is given now in the 7th revision of the 18th edition of the Kourovka Notebook (http://arxiv.org/abs/1401.0300):

This group is infinite, because it contains the Fibonacci group $F(3, 7)$ as an index $7$ subgroup. This follows from Theorem 3.0 of (C. P. Chalk, Commun. Algebra 26, no. 5 (1998), 1511–1546) by standard technique for working with Fibonacci groups (G. Williams, Letter of 6 October 2015).

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