(2k-l)a = w(2k-l).(2k-2), (2k-l)b = 2k, 2k b = w(2k).(2k-l), (2k)a = 2k+1,
where w(k) is given by w(1) = x, w(2) = y, w(3) = xy^x, w(4) = yx^^ and, by induction, w(k) = w(k-2)fw(k"3)]^; k > 5.
Each coset gives two relations for the subgroup <x,y>, one of the relations being trivial. Denote by R(k) = 1 the non-trivial relation obtained from coset k. Then from the relation ba^^bab ^a ^ = 1 we get
R(1) =* (xy”x)^yxy ^x \ 2n - 2 - 1
and from the relation ab aba b = 1 we get f 2n\n n -2 -1
R{2) ~ (yx ) xy xyx y . By induction,
R(k) = [ w(k+2)] ^w(k+1 )w(k) [w(k-1 )]' ^[w(k)] ^ ; k > 3.
Notice that if n < 1 the same argument holds. Thus we have the following theorem.
Theorem 3.4.1. If H is the subgroup <a,b^ of G(2,2,2n), then H S (x,y|R(k) = 1, k = 1,2,3, ...)
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where the R(k) are given inductively as above.
(Note that the fact that this is a presentation for H follows from Theorem 2,3.4, that is Theorem 4 of (2)).
We also have the following lemma.
Lemma 3.4.2. If H = < a,b^> is a subgroup of G(-1,-1,n) then H is normal in G(-1,-1,n) and G(-1,-1,n)/H is isomorphic to
Proof. We obtain a presentation for ÏÏ from the modified Todd-Coxeter algorithm. Define n cosets 1,2, ..., n by 1 = ( a,b^ > and ib = i+1, 1 < i < n - 1. From the subgroup generators y = b^ and x = a we obtain
^ —I "j ^ %% XI
nb = y1 and la = x1. Define w^ = x, w^ = x y x , w^ =
« • — "“IX. ♦
3 < < n. Then from the relation aba b a b a = 1 we obtain
2a = Wg2 and from the relation bab a b a b = l we obtain ia = w\i, 3 < i < n. The. relations for H come from considering
i.aba \ ^a \ ^a = R(i),i, 2 < i < n and 1.bab ^a \ ^a ^b = S(l),1, n.bab ^a ^b ^a % = S(n).n. Therefore H = < x,y I R(i) = 1,2 < i <n , S(l) = 1, S(n) = 1>, that is H=<x,y|yw^w^_j^^w^^ = 1 ;2 <i <n-1 ,yw^yw^y ^w^^^ = 1 , M — 1 •— 1 3.4.3. The group G(2,2,-3). / \ I 2 -"1 — 1 ^ —1 G(2,2,-3) = <a,b | ab a b a/b = 1 , ba b a b a = 1). Let H be the subgroup of G(2,2,-3) generated by [a \ b ^ ], [a \ b ], [ a,b ]. Clearly H < G'(2,2,-3) and in fact the coset enumeration programme (1) shows that {g(2 ,2,-3):H| = l8. Since by Lemma 3.2.6
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|g(2,2,“3) :G'(2,2,-3)| = 9, H is a subgroup of index 2 in G'(2,2,-3). Let X =[a \ b ^ ], y = [a \ b j and z = [ a,b ]. Then we may use the modified Todd-Coxeter algorithm to find a presentation for H, The
relations could be obtained by hand but we used the Wilde programme (4t), see 2.5, to find the words in the subgroup generators which give the relations between the coset representatives. We then used the programme CCRG, see 2.5, which obtained from these words a presentation for the subgroup H. The following twenty relations were obtained for H: —2 — 1 “ 1 2 —1 — 1 — 1 — 1 — 1 — 1 — 1 . X yz y X z y xzy xyzy x yz x yz = 1, — 1 —1 —1 — 1 2 — 1 — 1 — 1 — 1 — 1 2 , z y xzy xy x yx yzy x yz y x = 1 , — 1 — 1 — 1 — 1 —1 — 1 —1 — 1 — 1 — 1 —2 ” 1 , X yzy xy x yz x yzy x yz x yzx yzy = 1,
— 1 —1 2 —1 —1 — 1 —1 —1 —1 —1 —1 —1 2 —1 —1 —1 —1 —1 2 —1 —1 z y X z y xzy xyz y xy x yz y x z y x yz y x z y
— 1 —2 —1 —1 —1 —1 —1 —2 —1 —1—1 —1 —1 —2 — 1 — 1 xzy xyz x yzy x yz x yzx yx yzy x yz x yzx yzy xyz
” 1 2 — 1 — 1 — 1 — 1 — 1 — 1 —1 2 — 1 —1 — 1 — 1 — 1 — 1 —2 — 12 — 1 — 1 y x z y x y x y x y z y x z y xzy x yz x yzx yzy x z y
— 1 xzy X - 1,
— 1 — 1 — 1 2 —1 — 1 — 1 — 1 — 1 2 — 1 —1 —1 — 1 ~2 —1 2 — 1 — 1 — 1 X yz y X z y xzy xyz y xz y x yz x yzx yzy % z y xzy
— 1 —1 2 — 1 —1 — 1 — 1 —1 2 —1 — 1 —1 —1 —2 — 1 —2 —1 xz y X z y xzy xyz y xz y x yz x yzx yzy xyzx yzy = 1,
—2 — 1 — 1 2 — 1 — 1 — 1 — 1 — 1 — 1 2 — 1 — 1 — 1 — 1 — 1 2 — 1 — 1 — 1 X yz y X z y xzy xyz y xy x z y xzy xyz y xz y x yz
— 1 —2 — 1 — 1 - 1 — 1 2 — 1 —1 — 1 , X yzx yzy xyzx yz y x z y xzy xy = 1,
“ 1 “1 — 1 — 1 —2 — 1 —1 — 1 _ . .
yzx yx yz x yzx yx yz x yzx yz y x z y xzy xyz x yzx — 1 — 1 — i — 1 — 1 — 1 — 1 — 1 — 1 —2 —1 —1 —1 —1 ,