4 A. J. DUNCAN AND J. HOWIE

a'

FIGURE

1.

Corollary. If five

or

more of the eight corner

labels are nontrivial elements of G,

then the system of equations has a solution in

an overgroup of G.

Proof. If

five corners of a cube are coloured, then it is possible to choose a pair of

opposite faces, each of which has at least two coloured corners (try it!).

Proof of Theorem 1.

We

construct the solution group

H

of Theorem

1

in a series

of HNN extensions beginning from G.

Before beginning let us fix once and for all a

pair of opposite faces of the cube.

Without loss of generality, we choose the top and

bottom face (as in Figure 1). Thus we are assuming that

at least two of a, (3, /,

8

are nontrivial elements of G, and similarly for a', (3', 1', 8'.

Let Go

=

G

*

x

,

the HNN extension of

G

with

trivial associated subgroup. Note that each of the

elements axb, cx-1d has infinite order in G0

,

so we may form the HNN extension

G1

=

Go, y

I

(axb)y(cx- 1d)y-

1

of Go.

Lemma 2. If at least two

of a, (3, /,

8 are nontrivial elements of G, and similarly

for a',(3',!',8',

then the elements eyf,lx-1 i form

a

basis

for a

free

subgroup of

rank 2 in

G

1

,

and similarly

for

gy-1h,jxk.

We then form the HNN extension