# Problems from the Kourovka Notebook that undergraduate students can fully appreciate

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.

• I don't have such a list, but if you just want an example where the formulation of the problem may be understood from the definitions expected to be known to undergraduates, then what about, for example, problems like 8.10, 11.10, 16.3 ? Just randomly picked up three of them during random scrolling over the pdf. – Alexander Konovalov Dec 25 '14 at 19:30
• The two very first problems are easy to understand for anyone with a basic background in group theory. – Tobias Kildetoft Dec 25 '14 at 19:31
• 18.49 is another simply formulated example of a problem. – Alexander Konovalov Dec 25 '14 at 19:55
• I suggest to reformulate the title to better reflect what does "accessible to non-specialists" mean (i.e. "described in a way accessible to undegraduate maths students"). Then the question may give details on prerequisites they have (e.g. knowledge of definitions of a group, subgroup, conjugacy classes etc). I agree that there is a pedagogical value in knowing such examples from Kourovka, e.g. to say "this is a known theorem, and that is a slight variation of it which is an open problem" or to formulate an exercise like "check this conjecture for all finite groups of order up to N" etc. – Alexander Konovalov Dec 25 '14 at 20:38
• Also, what about problems 8.85, 15.83, 17.76, 18.56 ? – Alexander Konovalov Dec 26 '14 at 19:12

## 4 Answers

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).

• From 1982... is this still open? – Lopsy Feb 15 '15 at 16:40
• @Lopsy: yes, still open according to the 18th edition (2014), see page 27 there. BTW, the 1st edition was published in 1965, and as Preface to the 18th edition says, "Maybe the most striking illustration of its [i.e. Kourovka Notebook] success is the fact that more than $3/4$ of the problems from the first issue have now been solved." – Alexander Konovalov Feb 15 '15 at 17:14

Problem 15.99 from the 15th edition (2002):

Let $f(n)$ be the number of isomorphism classes of finite groups of order $n$. Is it true that the equation $f(n) = k$ has a solution for any positive integer $k$? The answer is affirmative for all $k \le 1000$ [G. M.Wei, Southeast Asian Bull. Math., 22, no. 1 (1998), 93–102]. (W. J. Shi)

Problem 17.76 from the 17th edition (2010):

Does there exist a finite group $G$, with $|G| > 2$, such that there is exactly one element in $G$ which is not a commutator? (D. MacHale)

Problem 18.49 from the 18th edition (2014):

Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying $1 < a, b, c \le n-2$ the symmetric group $S_n$ has elements of order $a$ and $b$ whose product has order $c$? (S. Kohl)

Update: the 7th revision of the 18th edition of the Kourovka Notebook (http://arxiv.org/abs/1401.0300) says that the answer is positive and refers to the preprint A note on the product of two permutations of prescribed orders by Joachim König. From the abstract:

We prove a conjecture by Stefan Kohl on the existence of triples of permutations of bounded degree with prescribed orders and product 1. This result leads to an existence result for covers of the complex projective line with bounded degree and prescribed ramification indices.