Minimal transitive degree

How few points can a group be shuffled on, in one orbit? The minimal faithful transitive degree μt(G) is the least m such that G embeds in Sₘ as a transitive subgroup — one orbit, every point reachable from every other.

One orbit is one coset space

A transitive action of G on m points is the action on the cosets of a point stabilizer H ≤ G, with m = [G : H]; its kernel is the normal core core(H) = ⋂g∈G gHg⁻¹. So the search runs over subgroups, not over actions:

μₜ(G) = min { [G : H]  :  H ≤ G,  core(H) = 1 }.

Such an H is called core-free. The trivial subgroup is always core-free (the regular action), so μt(G) is defined for every finite group, and:

• μt(G) divides |G| — it is the index of a subgroup (Lagrange).
• μt(G) ≤ |G|, with |G| attained by the regular action.
• μ(G) ≤ μt(G), where μ(G) drops the transitivity requirement.

When the regular action is the only option

μt(G) = |G| exactly when the trivial subgroup is the only core-free one — that is, when every nontrivial subgroup contains a nontrivial normal subgroup of G. Two ways this happens:

• Every subgroup is normal (Dedekind groups: all abelian groups, and the Hamiltonian ones such as Q₈). Then core(H) = H, so core-free means trivial.
• A unique minimal subgroup, which then lies in every nontrivial one — the cyclic p-groups Cpk and the generalized quaternion groups Q₂ⁿ. Q₁₆ is not Dedekind, yet its unique involution sits in every nontrivial subgroup, so μt(Q₁₆) = 16.

Contrast D₄, also of order 8: a reflection ⟨s⟩ is core-free of index 4, so μt(D₄) = 4.

Gap to the intransitive minimum

μ(G) = μt(G) whenever G is monolithic — it has a unique minimal normal subgroup. In a faithful family of orbits some member must escape that subgroup, and having escaped the only minimal normal subgroup its core is trivial, so that one orbit is already faithful.

When there are several minimal normal subgroups, splitting into orbits can pay. The extreme case is an abelian group, where μt = |G| always while μ is only the sum of the primary cyclic factors: μt(C₃ × C₃) = 9 against μ(C₃ × C₃) = 6, and μt(C₂ × C₃ × C₅) = 30 against μ = 10.

Values

• Dihedral. Every minimal normal subgroup of Dn (n ≥ 3) lies in the rotation subgroup ⟨ρ⟩, so a reflection avoids all of them; its core is trivial and its index is n. No smaller index is possible: a core-free subgroup meets the cyclic ⟨ρ⟩ trivially, hence has order at most 2.
• Dicyclic. Dicm has a unique involution, central and inside every even-order subgroup, and a core-free subgroup meets ⟨ρ⟩ trivially — leaving only the trivial subgroup. So the regular action of degree 4m is forced, as for the generalized quaternion groups Dic2k = Q2k+2.

See also

• Minimal faithful degree — the same question with several orbits allowed.
• Group action — orbits, stabilizers, and the normal core.
• Cayley's theorem — the regular action, degree |G|.
• Wikipedia: Permutation group