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Example text

Induced actions. Suppose that G acts on Ω. This one group action gives rise to several induced actions. Perhaps the simplest way to obtain an induced action is by restriction to a subgroup of G. There is also an induced action of G on each of its orbits. This induced action is transitive, so that if G acts semiregularly on Ω, then it is regular on orbits. Let N G. Then g(N a) = N (ga) for all g ∈ G and a ∈ Ω, implying that G acts on the set Φ of N -orbits in Ω. This induced action has kernel containing N , and so G/N acts on Φ.

If x fixes every element of Ω then it acts trivially. The set of such elements in G is called the kernel of the action; this normal subgroup of G is the kernel of the corresponding permutation representation. If the action has trivial kernel—so that the only element of G fixing every element of Ω is the identity—then we say that the action is faithful, and that G is a permutation group. If G is a permutation group on Ω then we sometimes do not distinguish it from its isomorphic image in Sym(Ω). When Ω is finite, G ≤ Sym(Ω) is finite.

Under this equivalence, the dual corresponds to H ∗ . It follows that H ∗ is a GH(n; G). Note: this proves that H ∗ is a GH(n; G) even when G is non-abelian. Since A∗ is a PCD(ΛGH(n;G) ) if and only if A is too, we say that ΛGH(n;G) is a conjugate transposable orthogonality set. We summarize the above deliberations as a theorem. 8. Theorem. Let G be a group of order dividing n. (1) ΛGH(n;G) is conjugate transposable. 22 2. MANY KINDS OF PAIRWISE COMBINATORIAL DESIGNS (2) ΛGH(n;G) is transposable if and only if G is abelian.

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