Problems from the Kourovka Notebook that undergraduate students can fully appreciate

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...

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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).

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 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)