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).
Best Answer
Stir a cup of tea, but don't let your spoon touch the walls of the cup. Once the tea has settled, there is some point in the liquid which is in exactly the same place in the cup as before you stirred it.
This is a consequence of the Brouwer Fixed Point Theorem.