BHASKAR RAO DESIGNS AND THE ALTERNATING GROUP A_{4}

D. COMBE, W.D. PALMER, AND W.R. UNGER

Abstract. In this paper we introduce a new construction for generalized Bhaskar Rao designs. Using this construction, we show that a generalized Bhaskar Rao design, GBRD(v, 3, λ; A_{4}) exists if and only if λ ≡ 0 (mod 12).

1. Introduction

In this paper we give a new construction for generalized Bhaskar Rao designs and then use this construction to establish a set of necessary and sufficient conditions for the existence of a generalized Bhaskar Rao design, GBRD(v, 3, λ; A_{4}). In particular, we show that a GBRD(v, 3, λ; A_{4}) exists if and only if λ ≡ 0 (mod 12).

The alternating group, A_{4}, is the group of even permutations on four letters and can be generated by means of the defining relations:

a^{3} = 1, b^{2} = c^{2} = d^{2} = 1, bc = d, ba = ad, ca = ab, da = ac.

2. Bhaskar Rao designs

Let 𝔾 be a finite group of order g, written multiplicatively. We write Z(𝔾) for the group ring of the group 𝔾 over the integers Z. Every element of Z(𝔾) is a formal sum ∑_{x∈𝔾} a_{x}x, where a_{x} is an integer and x is an element of the group 𝔾. The element ∑_{x∈𝔾} a_{x}x, where all a_{x} = 0, is called the zero element of Z(𝔾) and is denoted by 0. An element ∑_{x∈𝔾} a_{x}x of Z(𝔾) which has, for some x ∈ 𝔾, a_{x} = 1 and a_{y} = 0 for x ≠ y, is simply written as x and is said to be an element of Z(𝔾) lying in 𝔾 or a group element of Z(𝔾). As we will be concerned exclusively with group rings over the integers Z, we will often, when the meaning is clear, refer to the group element x lying in group ring Z(𝔾) informally as an element of the group 𝔾.

We define the element (∑_{x∈𝔾} a_{x}x)^{−1} of Z(𝔾) by

(∑_{x∈𝔾}a_{x}x)^{−1} = ∑_{x∈𝔾}a_{x}x^{−1}.

It follows that 0^{−1} = 0, the zero element of Z(𝔾), and, for any group element x, the group element x^{−1} is 1x^{−1}, a group element of Z(𝔾). For further information on group rings the reader is referred to, for example, Hall [6, pp. 255–261] or Aschbacher [1, pp. 35–42].

Date: 14 September 2000.

For a matrix A with entries in Z(𝔾), we define the size of a column of A to be the number of non-zero entries occurring in that column. If all columns of A have the same size, k (say), we call k the size of A.

Definition 1. Let 𝔾 be finite group of order g, written multiplicatively. We define a generalized Bhaskar Rao design GBRD(v, b, r, k, λ; 𝔾) to be a v × b matrix with entries from Z(𝔾), all of which are either 0 or group elements such that

1. each row contains exactly r group element entries and exactly b−r zero element entries;

2. each column contains exactly k group element entries and exactly v − k zero element entries;

3. for any pair of distinct rows (x_{1}, x_{2}, …, x_{b}) and (y_{1}, y_{2}, …, y_{b}) the list

x_{1}y_{1}^{g−1}, x_{2}y_{2}^{g−1}, …, x_{b}y_{b}^{g−1}

contains each group element of Z(𝔾) (that is, element of 𝔾) exactly λ/g times.

We note that λ ≡ 0 (mod g), k ≤ v and λ/g ≤ r ≤ b.

Example 2. A GBRD(5, 10, 6, 3, 3; Z_{3}) is given below:

A = [\begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & a & a^{2} & 0 & 0 & 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & a^{2} & a & 0 & a & a^{2} & 0 & 1 \\ 0 & 1 & 0 & a^{2} & 0 & a & 1 & 0 & a & a \\ 0 & 0 & 1 & 0 & a^{2} & a & 0 & a^{2} & 1 & a^{2} \end{matrix}]

The generalized Bhaskar Rao design, A, appearing in Example 2 comes from Seberry [11, p. 379]. The column size of A is 3 and each entry of A belongs to {1, a, a^{2}} ∪ {0}. For example, for the rows 2 and 3 of A, the list: 1, 0, 0, 0, 0, a^{−1} = a^{2}, (a^{2})^{−1} = a, 0, 0, contains each of the group elements 1, a and a^{2} exactly once.

Example 3. A GBRD(3, 12, 12, 3, 12; A_{4}) is given below:

[\begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & a & a^{2} & b & ab & a^{2}b & c & ac & a^{2}c & d & ad & a^{2}d \\ 1 & a^{2} & a & c & a^{2}c & ad & d & a^{2}d & ab & b & a^{2}b & ac \end{matrix}]

If v > k, replacing the group element entries in a GBRD(v, b, r, k, λ; 𝔾) produces a (0, 1)-matrix which is an incidence matrix for a balanced incomplete block design, BIBD(v, b, r, k, λ). It is well-known, see, for example, Street and Street [12], that the five numbers v, b, r, k and λ for a balanced incomplete block design, are not independent: they satisfy the relations bk = vr and λ(v − 1) = r(k − 1). For a generalized Bhaskar Rao design in which v = k, there are no zero element entries and b = r = λ, so the relations: bk = vr and λ(v − 1) = r(k − 1) still hold. Thus it is usual to denote the GBRD(v, b, r, k, λ; 𝔾) by the shorter notation GBRD(v, k, λ; 𝔾).

Some generalized Bhaskar Rao designs can be described by a set of GBRD initial blocks. See, for example, Seberry [11]. In Example 4, we explain this description by means of an example.

Example 4. A set of initial blocks (mod 3, A_{4}) for a GBRD(4, 3, 12; A_{4}) is given below:

(∞_{1}, 1_{a^{2}d}, 2_{b}), (∞_{1}, 1_{ab}, 2_{c}), (∞_{1}, 1_{a^{2}b}, 2_{a^{2}c}),
(∞_{1}, 1_{a}, 2_{ac}), (∞_{1}, 1_{1}, 2_{ad}), (∞_{1}, 1_{d}, 2_{a^{2}}),
(0_{1}, 1_{a}, 2_{a^{2}}), (0_{1}, 1_{ac}, 2_{ab}).

A GBRD(4, 3, 12; A_{4}) has 4 rows and 24 columns. We think of the rows of this 4 × 24 matrix as labelled ∞, 0, 1 and 2. The GBRD(4, 3, 12; A_{4}) is built up from eight 4 × 3 submatrices B_{1}, …, B_{8} which are developed from the eight initial blocks.

The collection of blocks developed (mod 3, A_{4}) from the initial block (∞_{1}, 1_{a^{2}d}, 2_{b}) is

(∞_{1}, 1_{a^{2}d}, 2_{b}), (∞_{1}, 2_{a^{2}d}, 0_{b}), (∞_{1}, 0_{a^{2}d}, 1_{b}).

That is, we develop an initial block, (a_{α}, b_{β}, c_{γ}), by developing a, b and c (mod 3) and leaving the subscripts α, β and γ unaltered.

From the three blocks:

(∞_{1}, 1_{a^{2}d}, 2_{b}), (∞_{1}, 2_{a^{2}d}, 0_{b}), (∞_{1}, 0_{a^{2}d}, 1_{b}),

we construct B_{1}. The entries of the first column of B_{1}, corresponding to the first block (∞_{1}, 1_{a^{2}d}, 2_{b}) are (1, 0, a^{2}d, b). That is, the group element 1 is placed in the row labelled ∞, a^{2}d in the row labelled 1, b is placed in row labelled 2, and the entry in row labelled 0 is 0, the zero element in Z(A_{4}).

Similarly, the entries of the second column of B_{1}, corresponding to the second block (∞_{1}, 2_{a^{2}d}, 0_{b}) are (1, b, 0, a^{2}d). Finally, the entries of the last column, corresponding to the third block (∞_{1}, 0_{a^{2}d}, 1_{b}) are (1, a^{2}d, b, 0).

That is,

B_{1} = [\begin{matrix}1 & 1 & 1 \\ 0 & b & a^{2}d \\ a^{2}d & 0 & b \\ b & a^{2}d & 0 \end{matrix}] .

Similarly, we construct the other seven matrices B_{2}, …, B_{8} each constructed from the blocks arising from developing the remaining seven initial blocks. The 4 × 24 matrix

B = [B_{1}, …, B_{8}]

is a GBRD(4, 3, 12; A_{4}). That is, we develop a collection of 24 blocks from the set of eight initial blocks. The GBRD(4, 3, 12; A_{4}) so constructed is given explicitly below:

[\begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & b & a^{2}d & 0 & c & ab & 0 & a^{2}c & a^{2}b & 0 & ac & a & 0 & ad & 1 & 0 & a^{2} & d & 1 & a^{2} & a & 1 & ab & ac \\ a^{2}d & 0 & b & ab & 0 & c & a^{2}b & 0 & a^{2}c & a & 0 & ac & 1 & 0 & ad & d & 0 & a^{2} & a & 1 & a^{2} & ac & 1 & ab \\ b & a^{2}d & 0 & c & ab & 0 & a^{2}c & a^{2}b & 0 & ac & a & 0 & ad & 1 & 0 & a^{2} & d & 0 & a^{2} & a & 1 & ab & ac & 1 \end{matrix}]

Example 5. A set of initial blocks (mod 5, A_{4}) for a GBRD(6, 3, 12; A_{4}) is given below:

\begin{matrix}(∞_{1}, 0_{1}, 3_{a^{2}}), & (0_{1}, 2_{a^{2}b}, 3_{a^{2}d}), & (∞_{1}, 0_{d}, 3_{a^{2}c}), & (0_{1}, 2_{c}, 3_{ac}), \\ (∞_{1}, 0_{b}, 3_{c}), & (0_{1}, 2_{a^{2}c}, 3_{1}), & (∞_{1}, 0_{ac}, 4_{a^{2}d}), & (0_{1}, 1_{1}, 4_{a}), \\ (∞_{1}, 0_{a}, 4_{ab}), & (0_{1}, 1_{ac}, 4_{ab}), & (∞_{1}, 0_{ad}, 4_{a^{2}b}), & (0_{1}, 1_{d}, 4_{a^{2}b}). \end{matrix}

3. Construction Theorems

Theorem 6. Let 𝔾 be a finite group of (order g), ℕ a normal subgroup (of order n) of 𝔾, and v, λ and μ positive integers. We write u for g/n, the order of 𝔾/ℕ and 0_{𝔾/ℕ} for the zero element in Z(𝔾/ℕ). Suppose we are given a v × b matrix, A, with entries taken from 𝔾/ℕ ∪ {0_{𝔾/ℕ}}such that

• for any pair of distinct rows (x_{1}, x_{2}, …, x_{b}) and (y_{1}, y_{2}, …, y_{b}), the list

x_{1}y_{1}^{u−1}, x_{2}y_{2}^{u−1}, …, x_{b} y_{b}^{u−1}

contains each element of 𝔾/ℕ exactly λ/u times.

• for each column size k of A, a GBRD(k, c, s, j, μ; ℕ), C(k). Then we can construct the matrix, X which is a GBRD(v, j, λμ; 𝔾).

Proof. Let t be the index of ℕ in 𝔾. Fix a set S = {g_{1} = e, …, g_{t}} of coset representatives of ℕ in 𝔾. We observe that the group element entries of the matrix A are cosets of ℕ in 𝔾 and the group element entries of each matrix C(k) are elements of the normal subgroup ℕ. For each k, we denote the k rows of C(k) by c(k)_{l}, l = 1, …, k.

We now form the matrix X from the matrices A and C(k). Select a column of A of size k (say). Replace each entry of this column by row vectors of length c in the following manner: replace the the first non-zero entry, say g_{l}ℕ by the row vector g_{l}c(k)_{1}, the second non-zero entry, say g_{m}ℕ by the row vector g_{m}c(k)_{2}, and so on. Finally, we replace the remaining (that is, zero entries) of the selected column of A by the row vector (0, …, 0) consisting of c zero entries.

Now select another column of A and repeat the replacement process.

When all columns of A have been replaced by row vectors we have constructed a matrix X which is a GBRD(v, j, λμ; 𝔾).

The following theorems which can be viewed as immediate consequences of Theorem 6 will be used extensively in the remaining sections of the paper. Theorem 7, based upon pairwise balanced designs (defined below), was first proved in de Launey and Seberry [5, 4], firstly for generalized Bhaskar Rao designs over the group Z_{2} and then for generalized Bhaskar Rao designs over any finite group.

For v and λ positive integers and K a set of positive integers. We define a pairwise balanced design, denoted by PBD(v; K; λ), to be an arrangement of the v elements of a set X into a collection (not necessarily distinct) subsets (called blocks) of X, for which:

1. each pair of distinct elements of X appear together in exactly λ blocks.

2. if a block contains exactly k elements of X then k belongs to K.

A pairwise balanced design PBD(v; {k}; λ), where K = {k} consists of exactly one integer, is a BIBD(v, k, λ). It is well-known, see, for example, Street and Wallis [13], that a PBD(v − 1; {k, k − 1}; λ) can be obtained from a BIBD(v, b, r, k, λ).

Theorem 8 was proved in Palmer [9].

Theorem 7. Given a pairwise balanced design PBD(v; K; λ), and for each k ∈ K, a GBRD(k, j, μ; 𝔾), we can construct a GBRD(v, j, λμ; 𝔾).

Proof. In Theorem 6, take 𝔾 = ℕ and A to be an incidence matrix of a PBD(v; K; λ).

Theorem 8. Suppose that ℕ is a normal subgroup of a finite group 𝔾. Then, given a GBRD(v, k, λ; 𝔾/ℕ) and a GBRD(k, j, μ; ℕ) we can construct a GBRD(v, j, λμ; 𝔾).

Proof. In Theorem 6, take A to be a GBRD(v, k, λ; 𝔾/ℕ).

In the next section, we apply Theorem 6 to construct a GBRD(14, 3, 12; A_{4}).

4. GBRD(14, 3, 12; A_{4})

The subgroup, ℕ = < b, c > is normal in A_{4} and is isomorphic to the group Z_{2} × Z_{2}. The factor group, A_{4}/ℕ is isomorphic to Z_{3}. The matrix, Y which is a GBRD(15, 7, 3; A_{4}/ℕ), found as a result of a Magma [3] search. We exhibit the matrix Y below:

[\begin{matrix}1ℕ & 0 & 0 & 0 & 0 & 1ℕ & 0 & 1ℕ & 0 & 0 & 1ℕ & 1ℕ & 0 & 1ℕ & 1ℕ \\ 1ℕ & 1ℕ & 0 & 0 & 0 & 0 & 1ℕ & 0 & 1ℕ & 1ℕ & 1ℕ & aℕ & 1ℕ & 0 & a^{2}ℕ \\ 1ℕ & aℕ & 1ℕ & 0 & 0 & 0 & 0 & aℕ & 0 & 1ℕ & 0 & 0 & a^{2}ℕ & a^{2}ℕ & 0 \\ 0 & 1ℕ & aℕ & 1ℕ & 0 & 0 & 0 & 0 & aℕ & 0 & 1ℕ & 0 & 0 & a^{2}ℕ & aℕ \\ 1ℕ & 0 & a^{2}ℕ & a^{2}ℕ & 1ℕ & 0 & 0 & 0 & 0 & aℕ & 0 & a^{2}ℕ & 0 & 0 & aℕ \\ 1ℕ & a^{2}ℕ & 0 & 1ℕ & a^{2}ℕ & a^{2}ℕ & 0 & 0 & 0 & 0 & aℕ & 0 & 1ℕ & 0 & 0 \\ 0 & 1ℕ & 1ℕ & 0 & a^{2}ℕ & aℕ & a^{2}ℕ & 0 & 0 & 0 & 0 & a^{2}ℕ & 0 & 1ℕ & 0 \\ 0 & 0 & 1ℕ & aℕ & 0 & a^{2}ℕ & aℕ & 1ℕ & 0 & 0 & 0 & 0 & 1ℕ & 0 & aℕ \\ 1ℕ & 0 & 0 & aℕ & aℕ & 0 & a^{2}ℕ & a^{2}ℕ & aℕ & 0 & 0 & 0 & 0 & 1ℕ & 0 \\ 0 & 1ℕ & 0 & 0 & 1ℕ & a^{2}ℕ & 0 & aℕ & a^{2}ℕ & aℕ & 0 & 0 & 0 & 0 & 1ℕ \\ 1ℕ & 0 & aℕ & 0 & 0 & aℕ & aℕ & 0 & a^{2}ℕ & a^{2}ℕ & a^{2}ℕ & 0 & 0 & 0 & 0 \\ 0 & 1ℕ & 0 & a^{2}ℕ & 0 & 0 & 1ℕ & a^{2}ℕ & 0 & 1ℕ & aℕ & 1ℕ & 0 & 0 & 0 \\ 0 & 0 & 1ℕ & 0 & 1ℕ & 0 & 0 & a^{2}ℕ & a^{2}ℕ & 0 & 1ℕ & aℕ & aℕ & 0 & 0 \\ 0 & 0 & 0 & 0 & 1ℕ & 1ℕ & 0 & 0 & a^{2}ℕ & 1ℕ & 0 & a^{2}ℕ & 1ℕ & aℕ & 0 \\ 0 & 0 & 0 & 0 & 1ℕ & 0 & a^{2}ℕ & 0 & 0 & a^{2}ℕ & aℕ & 0 & 1ℕ & a^{2}ℕ & 1ℕ \end{matrix}]

Now we delete the last row of Y to form A a 14 × 15 matrix. We observe that:

• the size of each column of A lies in {6, 7};

• for any pair of distinct rows of A, (x_{1}, x_{2}, …, x_{15}) and (y_{1}, y_{2}, …, y_{15}), the list

x_{1}y_{1}^{−1}, x_{2}y_{2}^{−1}, …, x_{15}y_{15}^{−1}

contains each element of the factor group A_{4}/ℕ exactly once.

The matrix C(7) which is a GBRD(7, 3, 4; ℕ), can be constructed by replacing the 1s in an incidence matrix for a BIBD(7, 3, 1) by the rows of D which is a GBRD(3, 3, 4; ℕ) where

D = [\begin{matrix}1 & 1 & 1 & 1 \\ 1 & b & bc & c \\ 1 & c & b & bc \end{matrix}] = [\begin{matrix}c & 1 \\ c & 2 \\ c & 3 \end{matrix}] .

The 0s are replaced by the zero row vector 0 = (0, 0, 0, 0). The matrix

[\begin{matrix}c_{1} & 0 & 0 & 0 & c_{1} & 0 & c_{1} \\ c_{1} & c_{1} & 0 & 0 & 0 & c_{1} & 0 \\ 0 & c_{2} & c_{1} & 0 & 0 & 0 & c_{2} \\ c_{3} & 0 & c_{2} & c_{1} & 0 & 0 & 0 \\ 0 & c_{3} & 0 & c_{2} & c_{2} & 0 & 0 \\ 0 & 0 & c_{3} & 0 & c_{3} & c_{2} & 0 \\ 0 & 0 & 0 & c_{3} & 0 & c_{3} & c_{3} \end{matrix}] = [\begin{matrix}e & 1 \\ e & 2 \\ e & 3 \\ e & 4 \\ e & 5 \\ e & 6 \\ e & 7 \end{matrix}] .

is then the required matrix C(7) which is a GBRD(7, 3, 4; ℕ).

From Lam and Seberry [8, p. 90], we see that the matrix

[\begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & b & c & bc & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & b & c & bc & 0 & 0 & 0 & b & c & bc & 0 & 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & b & 0 & 0 & c & bc & 0 & 1 & 0 & 0 & c & bc & 0 & c & bc & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & c & 0 & b & 0 & bc & 0 & 1 & 0 & bc & 0 & b & 0 & bc & b & c \\ 0 & 0 & 0 & 1 & 0 & 0 & bc & 0 & b & c & 0 & 0 & 1 & 0 & c & b & b & 0 & c & 1 \end{matrix}] = [\begin{matrix}f & 1 \\ f & 2 \\ f & 3 \\ f & 4 \\ f & 5 \\ f & 6 \end{matrix}] .

is a GBRD(6, 3, 4; ℕ). Set this general Bhaskar Rao design to be the matrix C(6). We now construct the matrix X from the entries of the matrix A and the rows of

the matrices C(6) and C(7) using the construction contained in Theorem 6

Each column of A is of size 6 or 7. Consider in turn the columns of A of size 6.

• Replace the first non-zero entry, say 1ℕ, by the row vector 1f_{1},

• Replace the second non-zero entry, say hℕ, where h ∈ {1, a, a^{2}}, by the row vector hf_{2}, where f_{2} is the second row of the matrix C(6); and so on.

• Replace the zero entries by a zero row vector of length 20. Next consider in turn the columns of A of size 7.

• Replace the first non-zero entry, 1ℕ, by the row vector 1e_{1}, where e_{1} is the first row of the matrix C(7).

• Replace the second non-zero entry, say hℕ, where h ∈ {1, a, a^{2}}, by the row vector he_{2}, where e_{2} is the second row of the matrix C(7); and so on.

• Replace the zero entries by a zero row vector of length 28.

The matrix X is a GBRD(14, 3, 12; A_{4}).

5. Existence result

Fix ℕ = <b, c>, a normal subgroup of A_{4}.

Lemma 9. Necessary conditions for the existence of a GBRD(v, 3, λ; A_{4}) are v ≥ 3 and λ ≡ 0 (mod 12).

Proof. For a GBRD(v, 3, λ; A_{4}) to exist

λ ≡ 0 (mod 12)

and there must exist a BIBD(v, 3, λ) which exists only if:

v ≥ 3
λ(v − 1) ≡ 0 (mod 2)
λv(v − 1) ≡ 0 (mod 6).

These conditions are equivalent to the necessary conditions:

v ≥ 3
λ ≡ 0 (mod 12)

From Hall [7, Lemma 15.4.2] we have the useful

Lemma 10. If v ≥ 3 then a PBD(v; K_{3}^{2}; 1) exists, where K_{3}^{2} = {3, 4, 5, 6, 8, 11, 14}.

We now construct a GBRD(u, 3, 12; A_{4}) where u ∈ {3, 4, 5, 6, 8, 11, 14}. Whence we apply Lemma 10 and Theorem 7 to construct a GBRD(v, 3, 12; A_{4}) for all v ≥ 3.

Theorem 11. If v ≥ 3 and odd. Then we can construct a GBRD(v, 3, 12; A_{4}).

Proof. In Seberry [11] it was shown that a GBRD(v, 3, 3; Z_{3}) exists when v ≥ 3 and odd. Hence there exists a GBRD(v, 3, 3; A_{4}/ℕ) Also a GBRD(3, 3, 4; Z_{2} × Z_{2}) exists (Lam and Seberry [8, Corollary 3.5.]) so a GBRD(3, 3, 4; ℕ). Hence, by Theorem 8, we can construct a GBRD(v, 3, 12; A_{4}) when v ≥ 3 and odd.

Corollary 12. A GBRD(v, 3, 12; A_{4}) exists for v ∈ {3, 5, 7, 11}.

Lemma 13. We can construct a GBRD(8, 3, 12; A_{4}).

Proof. GBRD(8, 4, 3; A_{4}/ℕ) exists as a GBRD(8, 4, 3; Z_{3}) exists (de Launey and Seberry [4]) A GBRD(4, 3, 4; ℕ) exists as a GBRD(4, 3, 4; Z_{2} × Z_{2}) exists (Lam and Seberry [8]). Hence, using Theorem 8, we can combine these designs to construct a GBRD(8, 3, 12; A_{4}).

Theorem 14. A generalized Bhaskar Rao design, GBRD(v, 3, λ; A_{4}) exists if and only if λ ≡ 0 (mod 12).

Proof. We have constructed a generalized Bhaskar Rao design, GBRD(u, 3, λ; A_{4}) for each u ∈ {3, 4, 5, 6, 8, 11, 14}. Hence, by Lemma 10 and Theorem 7, we can construct a GBRD(v, 3, 12; A_{4}) for all v ≥ 3. Finally, for all v ≥ 3 and for all λ = 12s, we can construct a GBRD(v, 3, λ; A_{4}) by taking s copies of a GBRD(v, 3, 12; A_{4}).

In Lemma 9 we proved that a GBRD(v, 3, λ; A_{4}) exists only if v ≥ 3 and λ ≡ 0 (mod 12).

References

[1] Aschbacher, M., Finite Group Theory, Cambridge University Press, Cambridge, 1986.

[2] Bhaskar Rao, M., Balanced orthogonal designs and their application in the construction of some BIB and group divisible designs, Sankhya Ser. A 32, 439–448, 1970.

[3] Bosma, W., Cannon, J.J., Playoust C., Steel, A., Solving Problems with Magma The School of Mathematics and Statistics, The University of Sydney, Sydney, 1995.

[4] de Launey, W. and Seberry, J., On generalized Bhaskar Rao designs of block size four, Congressus Numerantium, 41, 229–294, 1984.

[5] de Launey, W. and Seberry, J., On Bhaskar Rao designs of block size four, Combinatorics and Applications (Calcutta, 1982), 311–316, Indian Statist. Inst., Calcutta, 1984.

[6] Hall, M. Jr., The Theory of Groups, Macmillan, New York, 1959.

[7] Hall, M. Jr., Combinatorial Theory, 2nd edition, Wiley, New York, 1986.

[8] Lam, C. and Seberry, J., Generalized Bhaskar Rao designs, J. Statist. Plann. and Inference 10, 83–95, 1984.

[9] Palmer, W.D., A composition theorem for generalized Bhaskar Rao designs, Australas. J. Combin. 6, 221–228, 1992.

[10] Palmer, W.D. and Seberry, J., Bhaskar Rao designs over small groups, Ars Combinatoria 26A, 125–148, 1988.

[11] Seberry, J., Some families of partially balanced incomplete designs, In: Combinatorial Mathematics IX, Lecture Notes in Mathematics No. 952, Springer-Verlag, Berlin, 1982.

[12] Street, A.P. and Street, D.J., Combinatorics of Experimental Design, Oxford University Press, Oxford, 1987.

[13] Street, A.P. and Wallis, W.D., Combinatorial Theory: An Introduction, Charles Babbage Research Centre, Winnipeg, 1977.

D. Combe, School of Mathematics, The University of New South Wales, UNSW SYDNEY NSW 2052, Australia

E-mail address: diana@maths.unsw.edu.au
URL: http://www.maths.usyd.edu.au:8000/u/dianac/

W.D. Palmer, School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia

E-mail address: billp@maths.usyd.edu.au
URL: http://www.maths.usyd.edu.au:8000/u/billp/

W.R. Unger, School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia

E-mail address: billu@maths.usyd.edu.au