Internat. J. Math. & Math. Sci.
Vol. 8 No. 4 (1985) 821-824
821
ON MENNICKE GROUPS OF DEFICIENCY ZERO
MUHAMMAD A. ALBAR
Department of Mathematical Sciences
University of Petroleum and Minerals
Dhahran, Saudi Arabia
(Received larch 26, 1985 and in revised form May 20, 1985)
H(m,n,r)
The Mennicke group
ABSTRACT.
<x,y,z x
y
the few known 3-generator groups of deficiency zero.
x
m
y
z
y
n
z
r
z >
x
Several cases of
is one of
M(m,,r)
are
studied.
KEY WORDS AND PHRASES. Presentation, Reidemeister-Schreier method, relation matrix.
1980 AMS SUBJECT CLASSIFICATION CODE. 20F05.
Mennicke [i] has given a class of three generator three relation
M(m,n,r)
r e 3
<x,y,z x
y
x
m
y
z
y
(see also Higman [2].)
provided that neither
n2
n
defined by
which he proves to be finite for
m
n
nor
i,
r2
For general
m,n,r
the above
m,n,r
a)
The group
M’
is abelian and
M(3 3 3)
M
IMI
THEOREM 1.
PROOF.
groups
The aim of this paper is to consider the group for several cases with
r
general
z
r
Wamsley [3] discussed the group for some cases with
group is difficult to consider.
m
z
x
Macdonald [3] has shown that the above group is finite
i,
m
n
IMI
211
We notice that
<x,y,z[x y
divides
,
M
211
Z2
Z2
y
x
z
y
z
x
z >.
We use his result that
Wamsley has shown that
M’
is abellan and prove:
A straightforward application of the
Z2.
Reidemeister-Schreier rewriting process can be used to find the order of
suppress the details and merely notice that the relation matrix for
0
0
0
0
0
0
0
8
0
o
0
0
8
0
0
0
0
0
M’
M’
We
is
Z
211
Z8
Z and [M[ 23(23 23 22
Another group of deficiency zero is Johnson’s group
[4],
y
n-2
-1
n+2
z
r-2
-1
r+2
x
m-2 -1 m+2
J(m,n,r)
<x,y,z x
y
x
y
y
z
y
z
z
x
z
x
Therefore
M’
REMARK I.
The order of
J
J(2,2,2)
is 7.2
II,
[4].
A question could be raised here if
M
and
M.A. ALBAR
822
J
the 2-Sylow subgroup of
y
-i
2
z
z
-I
2
J
subgroup of
M
-r
b)
x
y
(m
x
x
m
y
and
n-1
PROOF:
M
Z
d
H
Therefore
7
-I
y
2
is the 2-Sylow
H
y
n-1
e>, m
=
M
Z(m
n-1
where
H
Zn_
m
d
x
M
Therefore
_
I)
n-I
m
<a
is (m
x
m-I >
The relations
H
M’
n-1
2.
I).
The relations
We consider
H
<x
is metacyclic and it is the split
a
But
2, n
>
x
and
a
y
a
a
-I y -I xy e
x
a
m
imply that
M’----H _c M’
M’.
m
The order of a is
n-I
(m-l,
REMARK 2.
m
x
is abelian implies that
H
Therefore
Z
<x
A student K. F. Lee of David L. Johnson showed
<x,ylx y
I),
by
We consider
M
H
n-
n-I
M’
THEOREM 2.
J
and
imply that the order of
e
Z(m
Z
Z2
M(m,n,0)
n-1
extension of
M
are different.
M
The group
Z
Z2
H
and
HJ
H
Using the Redemeister-Schreier process we write a presentation for
H
which gives
that
We find that
x > <J
To answer this question let
are isomorphic.
m
m
n-i
n-i
m
m-
i)
+ mn-3 +
n-2
+
m
2
+
m
+
The above theorem could be proved using the Reidemeister-Schreier process.
REMARK 3.
(m-i) (n-l)
’r
IMI
implies that
(n-l) (m
n-I
i)
REMARK 4.
The above theorem implies that
REMARK 5.
It is easy to see that
M(a,b,c)
REMARK 6.
M(a,c,b) in general.
In working with Mennicke’s group we find the commutator identity (known as
M
is a finite metabilian group.
-=
M(a, b, c)
M(b, c, a)
M(c, a, b)
and
the Witt identity)
quite helpful.
Ix, y, z]
c)
get
z2
[Ix,y], z]
y-2x-2
x
Ix,y]
and
zlx y
<x, y,
x2
The relations of
M(-1,-1,-i)
M
<x, y,
x
y
THEOREM 3.
e)
M(2
z x
M
zlx y
and
x
<x, y
zlx y
and
z
y
2
z
x
z
2
Using the Witt identity we
z y
y
x2
x I?
Therefore, M
Y
z
y-
M
to get
xy-2x-3
z
gives
Finally
x-I
We define
in any group.
xy
y
z
z
x
and
E.
z-
-
x2y2z
We substitute in
x
y
x2
M
y
imply that
z
z2
z
y2
x
Thus
e.
z
y2
e.
x
A
z2
Reidemeister-Schreier process gives that M’
z-ly -I. Therefore, we have proved:
is an infinite metabilian group.
2, -I)
z-lY-lz-2yz
z-lx -l
z
e
straightforward application of the
generated by
y
x 17
z
give
2
x, y
We use the relations of
e
y
to get
x
x
which together with z
y
e
This identity holds for any
Ix, z2][z, y2][y, x2]
and use
d)
x, y ][y, z,
M(2,2,2)
M
xy
z
zX][z,
[x y
Z
Z
z- >. Using the Witt identity we get
We use this relation together with
the relations of M to get
Y
z
823
MENNICKE GROUPS OF DEFICIENCY ZERO I
x
z -4.
Y
Y
M
<y,
z
f)
M(-I, -i, 0)
z2
(yZ) z
zly
x
z
Substituting in
y3
z-
e
The relation
(yz) 2
e>
S
<x, y,
zlxy
y
M’
Schreier process we get that
M
THEOREM 4.
H
Thus
H
g)
M
and
M
<xl>
K
Therefore
Using the Reidemeisterx 2 ".
M’
H
H
Let
as follows.
<x
But
21>
It is easy to see
y-lx-lyx
x
e
M’
H
M’
z_
z
x
<z2>
is the split extension of
M
Therefore
see [5].
zM
M
<zl> is
<xl> by <zl>
It is easy to see that
We also notice that
--K
Z x Z
(z2) x
x- Iz_
z2
M’
normal in
H
z
-2
xz.K
M
where the
z-2
and xz 2 x-l=
M’
M’
K
THEOREM 5.
h)
M’
Therefore,
z2
zlz x
<x,
action is given by
T,us
Z
Z2
-F
M’
M(1, 0, -1)
and
z
M
e>
Thus
e
metabilian group.
is an infinite
M
y2
(yz)
becomes
y
We notice that
e
is infinite cyclic generated
It is possible to find
REMARK 7.
that
x -l,
z
y
x
and so
e
z
we get
M
is an infinite metabilian group.
It is easy to show the following cases:
M(I, I, O)= Z
M(3, 2, O)
(iii)
Z
M(I, I, I)= Zx Z
M(I, 2, O)
M(I, O, O) Z
(v)
M(0, 0, 0)= M(2, 2, 0)= M(2, 0, 0)= E
(i)
(viii)
M(I, m, n)
(x)
M(1, -I, O)
(iv)
Zm_
is infinite because
Z
M(-m, 2, O)
M(2, 3, O)= S
m
2
for m
M(I m, n)
M"(1,m,n)
M(-m, 0, O)
(xi)
Z2
Zm+1,
(vi)
Z
Z
for n >
n-i
M(m, 2, 0)= M(m, 0, O)
(ix)
(xii)
Z
Z
M(I, n, 0)
(vii)
(ii)
is infinite.
Zm+1,
0
m
0
Mennicke’s group was a generalization of a group given by Higman [2].
Another generalization of Higman’s group was considered by Fluch [6] as
<a,b,clb-aaba
H
a
m
c
-8bc8
b
n
a-caY
c
r
then H
a
M(m, n, r)
8
Another generalization of Mennicke’s group was given by Post [7] as follows:
We notice that when
G(m,n,r,s,t)
ACKNOWLEDGEMENT.
<a,b,clabma -I
b
n
bc
rb-I
c
s
cac
-i
a
t
I thank Dr. D. L. Johnson for his useful comments on this paper.
I
also thank the University of Petroleum and Minerals for the support I get for conducting
research.
REFERENCES
i.
2.
MENNICKE, J. Einige endliche Gruppen mit drei Erzeugenden und drei Relationen,
Arch. Math., 10 (1959) 409-18.
HIGMAN, G. A finitely generated infinite simple group, J. London Math. Soc. 26
(1951), 61-64.
824
M.A. ALBAR
3.
WAMSLEY, J. W.
4.
JOHNSON, D. L.
5.
ALBAR, M. A.
6.
7
The deficiency of finite groups, A Ph.D. thesis, University of
Queensland, 1968.
A new class of 3-generator finite groups of deficiency zero, J.
London Math. Soc. 2 (1979), 59-61.
On presentation of group extensions, Communications in Algebra, 12
(1984) 2967-2975.
FLUCH, W. A generalized Higman group, Nederl. Akad. Wetensch. Inda$. Math., 44
(1982), 153-166.
POST, M. J. Finite three-generator groups with zero deficiency, Comm. Algebra,
6(1978), 1289-1296.