Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 39 (1998), No. 1, 19-30.

Generators of Order two for B_{n} and three of its Double Covers

John Brinkman

Department of Mathematics, Liverpool Hope University College Hope Park, Liverpool L16 8ND, England

1. Introduction

In this paper the minimum number of involutions required to generate either the Hyperoctrahederal Group B_{n} or three of its seven double covers are calculated. In addition explicit multiplications for the three particular double covers are introduced, along with specific generators, of order two, for each of the groups.

Definition 1.1. Denote by i(G) the minimum number of involutions required to generate a group G.

As a starting point we observe that if i(G) = 1 then G is cyclic and if i(G) = 2 then G is dihedral. Hence for any group of order greater than two that is not isomorphic to a dihedral group we have i(G) ≥ 3.

To considering our particular double covers of B_{n} we require the double cover of S_{n} that lifts a transposition of S_{n} to an element of order two. This has the following standard presentation. Generators are z, s_{1}, s_{2}, …, s_{n−1} with relations

z^{2} = 1, zs_{i} = s_{i}z, s_{i}^{2} = 1 for 1 ≤ i ≤ n − 1,
(s_{j}s_{j+1})^{3} = 1 for 1 ≤ j ≤ n − 2,
s_{k}s_{h} = zs_{h}s_{k} for | h − k |> 1 and 1 ≤ h, k ≤ n − 1.

We will consider B_{n} as the permutation wreath product C_{2}pwrS_{n} (see chapter 4 [3]). The Schur multiplier of B_{n}, for n > 3, is C_{2} × C_{2} × C_{2} [2] and it will be convenient to introduce here a notation that encompasses all seven double covers. We follow the usual indexing of

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1998 Heldermann Verlag

the triples (ζ, λ, μ) = (±1, ±1, ±1) to distinguish between the seven double covers. Then we have the following presentation for the double covers of B_{n}, dependent on a particular triple.

Generators are b_{1}, b_{2}, …, b_{n} with the relations,

b_{i}^{2} = 1 for 1 ≤ i ≤ n, (b_{j}b_{j+1})^{3} = 1 for 1 ≤ j ≤ n − 2,
(b_{i}b_{j})^{2} = ζ for 1 ≤ i, j ≤ n − 1 and | i − j |> 1,
(b_{i}b_{n})^{2} = λ for 1 ≤ i ≤ n − 2, and (b_{n−1}b_{n})^{4} = μ.

2. i(B_{n}^{(1,1,1)})

We shall refer to the elements of C_{2}pwrS_{n} as wreath permutations.

Definition 2.1. For \underline{c} = (c_{1}, c_{2}, …, c_{n}) ∈ (C_{2})^{n} and π ∈ S_{n}, let

(\underline{c})_{π} = \underline{c}_{π} = (c_{π^{−1}(1)}, c_{π^{−1}(2)}, …, c_{π^{−1}(n)}).

We now have elements of the form

(\underline{c}; π), (\underline{d}; κ) for \underline{c}, \underline{d} ∈ (C_{2})^{n} and π, κ ∈ S_{n},

with multiplication being defined as (\underline{c} ; π) · (\underline{d} ; κ) = (\underline{c} · \underline{d}_{π} ; πκ). The identity element of C_{2}pwrS_{n} is then (\underline{1} ; 1), where \underline{1} = (1, 1, …, 1) and an inverse for a typical wreath permutation (\underline{c}; π) is (\underline{c}; π)^{−1} = ((\underline{c}^{−1})_{π^{−1}} ; π^{−1}), where (\underline{c}^{−1})_{π^{−1}} = (\underline{c}_{π^{−1}})^{−1}. Associativity follows by noting that, for π, κ ∈ S_{n},

(\underline{c}_{π})_{κ} = \underline{c}_{κπ} and (\underline{c} · \underline{d})_{π} = \underline{c}_{π} · \underline{d}_{π}.

Lemma 2.2. A wreath permutation (\underline{c} ; π) ≠ (\underline{1} ; 1) ∈ B_{n} is an involution if and only if π^{2} = 1 and \underline{c}_{π} = \underline{c}.

Proof. Let π^{2} = 1 and \underline{c}_{π} = \underline{c}, then (\underline{c}; π)^{2} = (\underline{c} · \underline{c}_{π} ; π^{2}) = (\underline{1} ; 1). □

As we shall be considering the decomposition of the permutation π it will be convenient at this point to introduce some notation.

Definition 2.3. The following product of δ + 1 disjoint transpositions

(α, β)(α + 1, β − 1)(α + 2, β − 2) … (α + δ, β − δ),

will be denoted by T (α, α + δ, α + β). We will also use [ ]_{I} to denote the integer part of a number.

Theorem 2.4. For n > 3, the following three involutions generate B_{n}.

a = ((−1, 1, …, 1); T (2, [n/2]_{I}, n + 1)),
b = ((−1, 1, …, 1); T (2, [(n + 1)/2]_{I}, n + 2)) and
c = (\underline{1} ; (1, 2)).
Moreover, i(B_{n}) = 3.

Proof. The permutation T (2, [n/2]_{I}, n + 1) fixes (−1, 1, …, 1) thus

ab = (\underline{1} ; (2, 3, …, n)), hence abc = (\underline{1} ; (1, 2, …, n)) ,

and it is clear that ⟨abc, c⟩ is isomorphic to S_{n}. Thus ⟨abc, c⟩ will generate all the generators b_{i}, for 1 ≤ i ≤ n − 1, from our presentation for B_{n}. It only remains to show that a, b and c also generate b_{n}.

So consider the following wreath permutations, which are clearly elements of ⟨abc, c⟩, f = (\underline{1} ; T (2, [(n + 1)/2]_{I}, n + 2)) and g = (\underline{1} ; (1, n)). Then we may generate (gbf)^{2}bf = ((1, …, 1, −1); 1) = b_{n}. □

3. i(B_{n}^{(1,1,−1)})

To investigate involutions in B_{n}^{(1,1,−1)}, n ≥ 4, we consider an alternative presentation for B_{n}^{(ζ,λ,μ)} [1]. Here we have generators w_{1}, w_{2}, …, w_{n} and r_{1}, r_{2}, …, r_{n−1} with relations,

r_{i}^{2} = 1 for 1 ≤ i ≤ n − 1, w_{i}^{2} = 1 for 1 ≤ i ≤ n,
(r_{i}r_{j})^{2} = ζ for 1 ≤ i, j ≤ n − 1 and | i − j |> 1,
(r_{i}r_{i+1})^{3} = 1 for 1 ≤ i ≤ n − 2,
w_{i}w_{j} = μw_{j}w_{i} for 1 ≤ i, j ≤ n, i ≠ j,
w_{i}r_{i} = λr_{i}w_{i+1} for 1 ≤ i ≤ n − 1,
w_{j}r_{i} = λr_{i}w_{j} for 1 ≤ i, j ≤ n, and j ≠ i, i + 1.

Now consider the subgroup of B_{n}^{(1,1,−1)}, for n > 3, which is generated by w_{i}, for 1 ≤ i ≤ n. This group, which we denote N, has the following presentation.

⟨w_{1}, …, w_{n} : w_{i}^{2} = 1, 1 ≤ i ≤ n and (w_{i}w_{j})^{2} = −1 for i ≠ j⟩

The fact that N is normal in B_{n}^{(1,1,−1)} follows readily from the two relations

r_{i}^{−1}w_{i}r_{i} = w_{i+1} for 1 ≤ i ≤ n − 1,
r_{i}^{−1}w_{j}r_{i} = w_{j} for 1 ≤ i, j ≤ n and j ≠ i, i + 1.

Also the subgroup generated by r_{1}, r_{2}, …, r_{n−1} is just S_{n} (e.g. let r_{i} be the transposition (i, i + 1)). Hence as N ⋂ S_{n} = 1, B_{n}^{(1,1,−1)}, for n ≥ 4, is the semi-direct product of N by S_{n}.

Definition 3.1. Let π be any permutation in S_{n} and N the subgroup of B_{n}^{(1,1,−1)} as defined above. Also let θ_{π} be the automorphism of N, where for

\underline{w} = w_{i}w_{j} … w_{k} ∈ N , θ_{π}(\underline{w}) = w_{π^{−1}(i)}w_{π^{−1}(j)} … w_{π^{−1}(k)}.

Then we denote by ϕ_{π} the homomorphism from S_{n} to the automorphism group of N defined by

ϕ_{π}(\underline{w}) = ϕ(θ_{π}(\underline{w})) = w_{π^{−1}(i)}w_{π^{−1}(j)} … w_{π^{−1}(k)}.

As we are multiplying permutations in S_{n} from the left, the fact that ϕ_{π} is a homomorphism follows as ϕ_{π′π′′} = ϕ_{π′′}ϕ_{π′}.

Hence B_{n}^{(1,1,−1)}, for n ≥ 4, may be considered as the semi-direct product of N by S_{n} with, for \underline{w}_{1}, \underline{w}_{2} ∈ N and π_{1}, π_{2} ∈ S_{n}, multiplication given by

(\underline{w}_{1} ; π_{1}) · (\underline{w}_{2} ; π_{2}) = (\underline{w}_{1} · ϕ_{π_{1}}(\underline{w}_{2}) ; π_{1}π_{2}).

An inverse for (\underline{w}_{1} ; π_{1}) is (ϕ_{π_{1}^{−1}}(\underline{w}_{1}^{−1}) ; π_{1}^{−1}) and the identity element is (1 ; 1).

Lemma 3.2. Let g = (w_{i} ; π) be any element of B_{n}^{(1,1,−1)}, n > 3. Then g is an involution if either π is the identity, or ϕ_{π}(w_{i}) = w_{i} and π^{2} = 1.

Proof. g^{2} = (w_{i}ϕ_{π}(w_{i}) ; π^{2}) = (1 ; 1). □

Theorem 3.3. For n ≥ 4, i(B_{n}^{(1,1,−1)}) = 3.

Proof. As one or two involutions will not generate B_{n}^{(1,1,−1)}, n > 3, it suffices to show that three involutions do in fact generate the group.

First consider the cycle (1, 3, 4, …, n) ∈ S_{n}, we may write this as the product π′.π′′, where

1. for n = 4, let π′ = (1, 3) and π′′ = (1, 4),

2. for n = 5, let π′ = (1, 4) and π′′ = (1, 5)(3, 4),

3. for n ≥ 6 let

π′ = (1, n − 1).T (3, [n/2]_{I}, n + 1) and
π′′ = (1, n).T (3, [(n + 1)/2]_{I}, n + 2).

Now we prove that the following three group elements, each of which have order two, generate B_{n}^{(1,1,−1)}.

a = (1 ; (12)), b = (w_{2} ; π′) and c = (w_{2} ; π′′).

We have abc = (1 ; (1, 2, 3, 4, …, n)). Thus a and abc generate a subgroup isomorphic S_{n}. Hence we may generate all r_{i}, for 1 ≤ i ≤ n − 1, from our presentation, along with

b′ = (1 ; π′) and d = (1 ; (2, n)), so that d.b.b′.d = (w_{n} ; 1).

Now all the generators w_{i}, for 1 ≤ i ≤ n, may be obtained as follows

(w_{i} ; 1) = (1 ; (i, n)) · (w_{n} ; (i, n)). □

4. i(B_{n}^{(1,−1,1)})

For the double cover B_{n}^{(1,−1,1)}, we have the following presentation.

Generators are b_{1}, b_{2}, …, b_{n} and relations,

b_{i}^{2} = 1 for 1 ≤ i ≤ n, (b_{j}b_{j+1})^{3} = 1 for 1 ≤ j ≤ n − 2,
(b_{i}b_{j})^{2} = 1 for 1 ≤ i, j ≤ n − 1 and | i − j |> 1,
(b_{i}b_{n})^{2} = −1 for 1 ≤ i ≤ n − 2, and (b_{n−1}b_{n})^{4} = 1.

We use a multiplication similar to that used in Section 2 for B_{n}^{(1,1,1)}, but here we will be dependent on the product of two homomorphisms.

Definition 4.1. Let ψ be the usual sign homomorphism from S_{n} to Z_{2} which maps even permutations to 0 and odd permutations to 1.

Definition 4.2. Let ρ be the mapping from (C_{2})^{n} to Z_{2} which takes the values, for \underline{c} ∈ (C_{2})^{n},

ρ(\underline{c}) = 0 if \underline{c} has an even number of negative entries, or
ρ(\underline{c}) = 1 if \underline{c} has an odd number of negative entries.

The fact that the above mapping ρ is a homomorphism follows readily by noting that ρ(\underline{c}) + ρ(\underline{d}) = 1 if and only if ρ(\underline{c} · \underline{d}) = 1.

Note also that as a permutation, π say, acting on \underline{c} does not alter the number of negative coordinates of \underline{c}, we have ρ(\underline{c}) = ρ(\underline{c}_{π}).

Proposition 4.3. Let C_{2} wr_{λ}S_{n} be the set of elements of the form

(−1)^{s}(\underline{c}; π), for \underline{c} ∈ (C_{2})^{n}, π ∈ S_{n} and s = 1 or 0.

Then C_{2} wr_{λ}S_{n} is a group with multiplication defined by

(−1)^{s_{i}}(\underline{c}; π) · (−1)^{s_{j}}(\underline{d}; κ) = (−1)^{s_{i}+s_{j}+ψ(π)ρ(\underline{d})}(\underline{c} · \underline{d}_{π} ; πκ).

Moreover C_{2} wr_{λ}S_{n} is isomorphic to B_{n}^{(1,−1,1)}.

Proof. Clearly closure holds, the identity element is (1; 1) and a typical element

(−1)^{s}(\underline{c}; π) has an inverse (−1)^{s+ψ(π^{−1})ρ(\underline{c}_{π}^{−1})}(\underline{c}_{π^{−1}}^{−1} ; π^{−1}).

Associativity follows as (\underline{d} · \underline{e}_{κ})_{π} = \underline{d}_{π} · (\underline{e}_{κ})_{π}, (\underline{e}_{κ})_{π} = \underline{e}_{πκ} and ρ(\underline{c}) = ρ(\underline{c}_{π}). That B_{n}^{(1,−1,1)} is embedded in C_{2} wr_{λ}S_{n} follows by letting

b_{1} = (\underline{1} ; (1, 2)), b_{2} = (\underline{1} ; (2, 3)), …, b_{n−1} = (\underline{1} ; (n − 1, n)),
b_{n} = ((1, 1, …, 1, −1); 1) and λ = −(\underline{1} ; 1).

The fact that we have an isomorphism now follows by noting that the order of B_{n}^{(1,−1,1)} is 2^{n+1}.n!, which is the same as that of C_{2} wr_{λ}S_{n}. □

Henceforth we shall refer to the elements of B_{n}^{(1,−1,1)} as signed wreath permutations in C_{2} wr_{λ} S_{n}. Thus we have a group multiplication similar to that of B_{n} but we must take account of the sign associated with the wreath permutation when determining involutions in B_{n}^{(1,−1,1)}.

Lemma 4.4. A signed wreath permutation

±(\underline{c}; π) ≠ (\underline{1} ; 1) ∈ B_{n}^{(1,−1,1)},

is an involution if and only if π^{2} = 1, \underline{c}_{π} = \underline{c} and either ψ(π) = 0 or ρ(\underline{c}) = 0.

Proof. Let π^{2} = 1, \underline{c}_{π} = \underline{c} and either ψ(π) = 0 or ρ(\underline{c}) = 0. then

((−1)^{s}(\underline{c}; π))^{2} = (−1)^{s}(\underline{c}; π) · (−1)^{s}(\underline{c}; π)
= (−1)^{2s+ψ(π)ρ(\underline{c})}(\underline{c} · \underline{c}_{π} ; π^{2})
= (−1)^{2s+0}(\underline{1} ; 1) = (\underline{1} ; 1). □

It will be convenient at this point to introduce some more notation. Let \underline{c}_{j} ∈ (C_{2})^{n} be such that the j^{th} component, c_{j} of \underline{c}_{j}, is negative, with c_{i} strictly positive for all i ≠ j.

Proposition 4.5. The following four involutions generate B_{4}^{(1,−1,1)}.

b_{1} = (\underline{1} ; (1, 2)), b_{2} = (\underline{1} ; (2, 3)), b_{3} = (\underline{1} ; (3, 4)) b_{4} = (\underline{c}_{4} ; 1) .

Moreover i(B_{4}^{(1,−1,1)}) = 4.

Proof. As the four given involutions satisfy our presentation for B_{n}^{(1,1,−1)}, we have i(B_{n}^{(1,1,−1)}) ≤ 4. The fact that i(B_{n}^{(1,1,−1)}) ≥ 4 follows as at least one of our generators must contain an odd number of negative c_{i} and, as we require the generator to have order two, this generator must have π = 1. Now each of our remaining generators must contain only permutations that have order two and also generate S_{4}, but i(S_{4}) = 3. □

Proposition 4.6. B_{5}^{(1,−1,1)} is generated by the following three involutions.

a = (\underline{c}_{1} ; (2, 5)(3, 4)), b = (\underline{1} ; (3, 5)), c = (\underline{1} ; (1, 2)) .

Proof. We only need show that a, b and c generate all the signed wreath permutations of the form (\underline{1} ; π) for any π ∈ S_{5}. As then we may generate

(\underline{1} ; (1, 5)) a (\underline{1} ; (2, 5)(3, 4))(\underline{1} ; (1, 5)) = (\underline{c}_{5} ; 1))
and ((\underline{c}_{5} ; 1)(\underline{1} ; (1, 2)))^{2} = −(\underline{1} ; 1)

thus all of B_{5}^{(1,−1,1)}. But as aba = (\underline{1} ; (2, 4)(3, 5)) it is readily verified that b, aba and c generate all the signed wreath permutations of the form (\underline{1} ; π). □

Proposition 4.7. For n ≥ 4, if n ≡ 2 modulo 4 then the following three involutions generate B_{n}^{(1,−1,1)}.

a = (\underline{c}_{1} ; T (2, [n/2]_{I}, n + 1)),
b = (\underline{c}_{1} ; T (2, [(n + 1)/2]_{I}, n + 2)) and
c = (\underline{1} ; (1, 2)).

Proof. From the proof of Theorem 2.4 it is clear that a, b and c will generate at least one, up to parity of sign, of each type of signed wreath permutation of the group. Thus we need only show that λ = −(\underline{1} ; 1) has also been generated as multiplication by λ will generate the signed wreath permutations of opposite parity. But we have generated either ±(\underline{c}_{n} ; (1, 2)) which squares to give λ. □

Proposition 4.8. When n = 7, 8 and 9 the following three involutions, dependent on n, generate B_{n}^{(1,−1,1)}.

n = 7, a = (\underline{1} ; (4, 5)(3, 6)), b = (\underline{c}_{7} ; (2, 5)(3, 4)) and c = (\underline{1} ; (1, 2)(4, 7)(3, 6)). n = 8, a = (\underline{1} ; (1, 2)(3, 5)(7, 8)), b = (\underline{c}_{6} ; (2, 4)(1, 5)) and c = (\underline{1} ; (3, 6)(5, 7)). n = 9, a = (\underline{1} ; (1, 2)(3, 6)(4, 5)(7, 8)), b = (\underline{c}_{9} ; (1, 3)(4, 6)) and c = (\underline{1} ; (6, 7)(8, 9)(3, 4)).

Proof. For n = 7 we have (ab)^{2} = ±(\underline{1} ; (2, 3, 4, 5, 6)),

(ac)^{3} = ±(\underline{1} ; (1, 2)) and (ab)^{2}.(ac)^{3} = ±(\underline{1} ; (1, 2, 3, 4, 5, 6)).

So we may generate d = ±(\underline{1} ; (1, 4)) and d′ = c.d.c = ±(\underline{c}_{1} ; (1, 7)) from which we obtain ±(\underline{1} ; (1, 2, 3, 4, 5, 6, 7)). The proof now follows using steps similar to those of Proposition 4.6. When n = 8 the proof is similar to that of n = 7 and is omitted. The proof for n = 9 requires more involved calculations, as we show. Here

(ab)^{2} = ±(\underline{1} ; (1, 3, 5)(2, 4, 6)) and (ac)^{7} = ±(\underline{1} ; (1, 2)).

Thus

(ab)^{2}. ± (\underline{1} ; (1, 2)).(ab)^{−2} = ±(\underline{1} ; (5, 6))
(ab)^{2}. ± (\underline{1} ; (5, 6)).(ab)^{−2} = ±(\underline{1} ; (3, 4))
b. ± (\underline{1} ; (5, 6)).b  = ±(\underline{1} ; (4, 5))
(ab)^{2}. ± (\underline{1} ; (4, 5)).(ab)^{−2} = ±(\underline{1} ; (2, 3))

and we have the required generators for the subgroup of B_{9}^{(1,−1,1)} isomorphic to S_{6}. Thus we may generate

x = ±(\underline{1} ; (1, 4)), y = ±(\underline{1} ; (1, 6)) and z = ±(\underline{1} ; (1, 3)).

Also we have

x′ = (ac). ± (\underline{1} ; (5, 6)).(ac)^{−1} = ±(\underline{1} ; (4, 8))
y′ = (ac). ± (\underline{1} ; (4, 8)).(ac)^{−1} = ±(\underline{1} ; (6, 9))
z′ = (ab)^{2}. ± (\underline{1} ; (4, 8)).(ab)^{−2} = ±(\underline{1} ; (3, 7))

which as

x′.x.x′ = ±(\underline{1} ; (1, 8)), y′.y.y′ = ±(\underline{1} ; (1, 9)), z′.z.z′ = ±(\underline{1} ; (1, 7)),

means that we may generate all the subgroup of B_{9}^{(1,−1,1)} isomorphic to S_{9} from which the proof follows as in the previous cases. □

Proposition 4.9. For n ≥ 11, the following three involutions, dependent on the value of n modulo 4, generate B_{n}^{(1,−1,1)}. n ≡ 0 (mod 4) a = (\underline{1} ; T (3, n/2, n + 1)),

b = (\underline{c}_{n} ; (2, 3)T (4, n/2, n + 2)) and

c = (\underline{1} ; (1, 2)(3, n)(4, n − 1)T (5, n/2, n + 1)).

n ≡ 1 (mod 4) a = (\underline{1} ; T (3, (n − 1)/2, n)), b = (\underline{c}_{n} ; (2, 3)T (4, (n − 1)/2, n + 1)) and

c = (\underline{1} ; (1, 2)(3, n)(4, n − 1)(5, n − 2)T (6, (n − 1)/2, n)).

n ≡ 3 (mod 4) a = (\underline{1} ; T (3, (n + 1)/2, n + 2)), b = (\underline{c}_{n} ; (2, 3)T (4, (n + 1)/2, n + 3)) and

c = (\underline{1} ; (1, 2)(3, n)T (4, (n + 1)/2, n + 2)).

Proof. For each n, the permutations in a and b fix \underline{c}_{n} and

a.b = ±(\underline{c}_{n} ; (2, 3, …, k))

thus a.b contains a permutation of length k − 1 where

k = \{ \begin{matrix}n − 2 & when n ≡ 0 (mod 4), \\ n − 3 & when n ≡ 1 (mod 4), \\ n − 1 & when n ≡ 3 (mod 4). \end{matrix}

Thus as

(ab)^{2} = ±(\underline{1} ; (2, 4, …, k, 3, …, k − 1))

it follows that

(ab)^{k} = ±(\underline{1} ; (2, 3, …, k)).

Also, for each n,

(ac)^{3} = ±(\underline{1} ; (1, 2)).

Thus consider when n ≡ 0, as in the previous proposition, we may generate either ±(\underline{c}_{n}; (1, 2)) which squares to give λ. So we may generate all the group B_{n−2}^{(1,−1,1)}. Hence we may generate d = ±(\underline{1} ; (2, 3)) and e = ±(\underline{1} ; (2, 4)) to obtain

d′ = d.c.d = ±(\underline{1} ; (1, n)) and e′ = e.d.e = ±(\underline{1} ; (1, n − 1))

from which all B_{n}^{(1,−1,1)} may be generated.

The proofs for n ≡ 1 and 3 modulo 4 follow similar steps. □

Theorem 4.10. For n ≥ 5, i(B_{n}^{(1,−1,1)}) = 3.

Proof. This is immediate from the previous propositions. □

5. I(B_{n}^{(1,−1,−1)})

We adapt the multiplication of Sections 3 and 4 for B_{n}^{(1,−1,−1)}, n > 3. It will be convenient to consider the following presentation of the group B_{n}^{(1,−1,−1)}. Take as generators w_{1}, w_{2}, …, w_{n} and r_{1}, r_{2}, …, r_{n−1} with relations,

r_{i}^{2} = 1 for 1 ≤ i ≤ n − 1, w_{i}^{2} = 1 for 1 ≤ i ≤ n,
(r_{i}r_{j})^{2} = 1 for 1 ≤ i, j ≤ n − 1 and | i − j |> 1,
(r_{i}r_{i+1})^{3} = 1 for 1 ≤ i ≤ n − 2,
w_{i}w_{j} = −w_{j}w_{i} for 1 ≤ i, j ≤ n,
w_{i}r_{i} = −r_{i}w_{i+1} for 1 ≤ i ≤ n − 1,
w_{j}r_{i} = −r_{i}w_{j} for 1 ≤ i, j ≤ n, and j ≠ i, i + 1.

In addition to the subgroup S_{n} of B_{n}^{(1,−1,−1)} we also have a subgroup A generated by w_{1}, w_{2}, …, w_{n}. This subgroup has the following presentation.

⟨w_{1}, w_{2}, …, w_{n} : w_{i}^{2} = 1, ≤ i ≤ n and (w_{i}w_{j})^{2} = −1 for i ≠ j⟩

Definition 5.1. Let

\underline{w} = ±w_{a_{1}}^{s_{a_{1}}}w_{a_{2}}^{s_{a_{2}}} … w_{a_{n}}^{s_{an}}, for s_{a_{i}} = 0 or 1,

where a_{i} is any positive integer from 1 to n, be any element of A. Then denote by k the number of non-zero s_{a_{i}} of \underline{w} and define by τ a map from A to Z_{2} which takes the values, for \underline{w} ∈ A,

τ(\underline{w}) = 0 if k is even and τ(\underline{w}) = 1 if k is odd.

As the mapping τ from A to Z_{2}, defined above, takes only the values 0 or 1 it is a straight forward exercise to show that τ is a homomorphism by noting that τ(\underline{w}) + τ(\underline{w}′) = 1 if and only if τ(\underline{w}\underline{w}′) = 1.

Definition 5.2. Let \underline{w} = w_{i}w_{j} … w_{k} be any word in A and π a permutation in S_{n}. Then we denote by ϕ_{π} the homomorphism from S_{n} to AUT A, the automorphism group of A, defined by

ϕ_{π}(\underline{w}) = w_{π^{−1}(i)}w_{π^{−1}(j)} … w_{π^{−1}(k)}.

Note that a permutation acting on \underline{w} does not alter the value of τ as only the positioning of the w_{i} ∈ \underline{w} may be altered, not the number of non-zero w_{i}.

Proposition 5.3. Let A wr_{λ}S_{n} be the set of elements of the form

(\underline{w}; π), for \underline{w} ∈ A, π ∈ S_{n}.

Then A wr_{λ}S_{n} is a group with multiplication defined by

(\underline{w}_{i} ; π) · (\underline{w}_{j} ; κ) = ((−1)^{ψ(π)τ(\underline{w}_{j})}\underline{w}_{i} · ϕ_{π}(\underline{w}_{j}) ; πκ).

Moreover for n ≥ 4, B_{n}^{(1,−1,−1)} is isomorphic to A wr_{λ}S_{n} Proof. Clearly closure holds. The identity element is (1; 1) and a typical element

(\underline{w}; π) has an inverse ((−1)^{ψ(π^{−1})τ(\underline{w}^{−1})}ϕ_{π^{−1}}(\underline{w}^{−1}) ; π^{−1}).

For associativity, as ϕ_{πκ} = ϕ_{π}ϕ_{κ} and τ(ϕ_{π_{j}}(\underline{w}_{k})) = τ(\underline{w}_{k}) we have

((\underline{w}_{i} ; π) · (\underline{w}_{j} ; κ)) · (\underline{w}_{k} ; δ)
= ((−1)^{ψ(π)τ(\underline{w}_{j})}\underline{w}_{i} · ϕ_{π}(\underline{w}_{j}) ; πκ) · (\underline{w}_{k} ; δ)
= ((−1)^{ψ(πκ)τ(\underline{w}_{k})}(−1)^{ψ(π)τ(\underline{w}_{j})}\underline{w}_{i}ϕ_{π}(\underline{w}_{j}) · ϕ_{πκ}(\underline{w}_{k}) ; πκδ)
= ((−1)^{ψ(π)τ(\underline{w}_{k})+ψ(κ)τ(\underline{w}_{k})+ψ(π)τ(\underline{w}_{j})}\underline{w}_{i}ϕ_{π}(\underline{w}_{j}) · ϕ_{πκ}(\underline{w}_{k}) ; πκδ)
= ((−1)^{ψ(π)τ(\underline{w}_{j})+ψ(π)τ(ϕ_{κ}(\underline{w}_{k}))+ψ(κ)τ(\underline{w}_{k})}\underline{w}_{i}ϕ_{π}(\underline{w}_{j}) · ϕ_{πκ}(\underline{w}_{k}) ; πκδ)
= ((−1)^{ψ(π)τ(\underline{w}_{j}ϕ_{κ}(\underline{w}_{k}))+ψ(κ)τ(\underline{w}_{k})}\underline{w}_{i}ϕ_{π}(\underline{w}_{j} · ϕ_{κ}(\underline{w}_{k})) ; πκδ)
= ((−1)^{ψ(π)τ((−1)^{ψ(κ)τ(\underline{w}_{k})}\underline{w}_{j}ϕ_{κ}(\underline{w}_{k}))}\underline{w}_{i}ϕ_{π}((−1)^{ψ(κ)τ(\underline{w}_{k})}\underline{w}_{j} · ϕ_{κ}(\underline{w}_{k})) ; πκδ)
= (\underline{w}_{i} ; π) · ((−1)^{ψ(κ)τ(\underline{w}_{k})}\underline{w}_{j} · ϕ_{κ}(\underline{w}_{k}) ; κδ))
= (\underline{w}_{i} ; π) · ((\underline{w}_{j} ; κ) · (\underline{w}_{k} ; δ))

Clearly the orders of both groups are the same as the number of elements of the form (\underline{w}; π) is 2^{n+1}.n!, as in B_{n}^{(1,−1,1)}. Thus A wr_{λ}S_{n} has 2^{n+1}.n! group elements, which is also the order of B_{n}^{(1,−1,−1)}. To see that B_{n}^{(1,−1,−1)} is embedded in A wr_{λ}S_{n}, identify −1 with (−1 ; 1), r_{i} with (1 ; (i, i + 1)) and y_{j} with (w_{j} ; 1). □

Hence we consider the elements of B_{n}^{(1,−1,−1)} as being of the form

(\underline{w}; π), for \underline{w} ∈ A, π ∈ S_{n}.

Lemma 5.4. Let (w_{i} ; π) be any element of B_{n}^{(1,−1,−1)}, n > 3. Then (w_{i} ; π) is an involution when either π is the identity, or ϕ_{π}(w_{i}) = w_{i} and also π has cycle type 2^{r} with r congruent to 0 or 2 modulo 4.

Proof. The statement of this proposition is similar to that of Lemma 3.2, the only difference being that we have imposed the extra condition that as τ(\underline{w}) ≠ 0 we have an even number of transpositions so that ψ(π) = 0. □

Proposition 5.5. The following four involutions generate B_{4}^{(1,−1,−1)}.

b_{1} = (\underline{1} ; (1, 2)), b_{2} = (\underline{1} ; (2, 3)), b_{3} = (\underline{1} ; (3, 4)), b_{4} = (w_{4} ; 1).

Moreover i(B_{4}^{(1,−1,−1)}) = 4.

Proof. This proof just follows the steps of Proposition 4.5 with \underline{c}_{4} replaced by w_{4}. □

Proposition 5.6. B_{5}^{(1,−1,−1)} is generated by the following three involutions.

a = (w_{1} ; (2, 5)(3, 4)), b = (\underline{1} ; (3, 5)), c = (\underline{1} ; (1, 2)) .

Proof. Again we need only refer to the proof of Proposition 4.6 with \underline{c}_{1} replaced by w_{1} and the result follows. □

Proposition 5.7. For n ≥ 4, if n ≡ 2 modulo 4 then the following three involutions generate B_{n}^{(1,−1,−1)}.

a = (w_{1} ; T (2, [n/2]_{I}, n + 1)),
b = (w_{1} ; T (2, [(n + 1)/2]_{I}, n + 2)) and c = (1 ; (1, 2)).

Proof. Note first that for each n ≡ 2 modulo 4 we have

ψ(T (2, [n/2]_{I}, n + 1)) = ψ(T (2, [(n + 1)/2]_{I}, n + 2)) = 0.

Thus it is clear by Theorem 2.4, with (−1, 1, …, 1) replaced by w_{1}, that a, b and c will generate a subgroup of B_{n}^{(1,−1,−1)} isomorphic to S_{n}. Hence we may generate

b′ = (1 ; T (2, [(n + 1)/2]_{I}, n + 2)) so that (c.b.b′)^{4} = (−1 ; 1).

Then as, for any 1 ≤ i ≤ n,

(w_{i} ; 1) = (−1 ; 1) · (1 (1, i)) · (w_{1} ; 1) · (1 (1, i))

we may generate all the group. □

Proposition 5.8. When n = 7, 8 and 9 the following three involutions, dependent on n, generate B_{n}^{(1,−1,−1)}. n = 7 a = (1 ; (4, 5)(3, 6)), b = (w_{7} ; (2, 5)(3, 4)) and c = (1 ; (1, 2)(4, 7)(3, 6)). n = 8 a = (1 ; (1, 2)(3, 5)(7, 8)), b = (w_{6} ; (2, 4)(1, 5)) and c = (1 ; (3, 6)(5, 7)). n = 9 a = (1 ; (1, 2)(3, 6)(4, 5)(7, 8)), b = (w_{9} ; (1, 3)(4, 6)) and c = (1 ; (6, 7)(8, 9)(3, 4)).

Proof. If we replace \underline{c}_{i} by w_{i} in each generator c, then Proposition 4.8 shows that for each n we may generate a subgroup of B_{n}^{(1,−1,−1)} isomorphic to S_{n}. The remainder of the proof now follows as in Proposition 5.7. □

Proposition 5.9. For n ≥ 11, the following three involutions, dependent on the value of n modulo 4, generate B_{n}^{(1,−1,−1)}. n ≡ 0 (mod 4) a = (1 ; T (3, n/2, n + 1)),

b = (w_{n} ; (2, 3)T (4, n/2, n + 2)) and

c = (1 ; (1, 2)(3, n)(4, n − 1)T (5, n/2, n + 1)).

n ≡ 1 (mod 4) a = (1 ; T (3, (n − 1)/2, n)), b = (w_{n} ; (2, 3)T (4, (n − 1)/2, n + 1)) and

c = (1 ; (1, 2)(3, n)(4, n − 1)(5, n − 2)T (6, (n − 1)/2, n)).

n ≡ 3 (mod 4) a = (1 ; T (3, (n + 1)/2, n + 2)), b = (w_{n} ; (2, 3)T (4, (n + 1)/2, n + 3)) and

c = (1 ; (1, 2)(3, n)T (4, (n + 1)/2, n + 2)).

Proof. First we note that, for each n, the permutation π in b is such that ϕ_{π}(w_{n}) = w_{n} and ψ(π) = 0, thus a, b and c are involutions.

As in Proposition 5.5 it suffices to show that for each n we may generate

(1 ; (1, 2)) and (1 ; (1, 2, …, n)).

But this follows easily from Proposition 4.9, with \underline{c}_{n} replaced by w_{n}. □

Theorem 5.10. For n ≥ 5, i(B_{n}^{(1,−1,−1)}) = 3.

Proof. This is immediate from the previous propositions. □

Acknowledgements

This article is an extension of part of a thesis which the author undertook while being supported by the SERC and supervised by Dr. J. F. Humphreys, to whom the author is greatly indebted, at Liverpool University.

References

[1] Davies, J. W.; Morris, A. O.: The Schur multiplier of the generalized symmetric group. J. London Math. Soc. (2) 8 (1974), 615–620.

[2] Ihara, S.; Yokonuma, T.: On the second cohomology groups (Schur Multipliers) of finite reflection groups. J. Fac. Sci. Univ. Tokyo, Sect. 1, X1 (1965), 155–171.

[3] James, G. D.; Kerber, A.: The Representation Theory of the Symmetric Group. Addison-Wesley, Reading (Mass.) 1981.

Received November 21, 1996