In this paper we study the Seiberg–Witten invariants of 4-manifold acted on by a finite group (or a compact Lie group). Among other things, we have: Let X be a smooth closed 4-dimensional ℤp-manifold, where p is a prime. Suppose H1(X,ℝ) = 0 and [Formula: see text] where [Formula: see text]. If ℤp acts trivially on the space [Formula: see text] of self dual harmonic 2-forms, then, for any ℤp-equivariant Spin c-structure [Formula: see text] on X, the Seiberg–Witten invariant satisfies [Formula: see text] provided [Formula: see text] for j = 0,1,…,p-1, where [Formula: see text], DA: Γ(W+) → Γ (W-) is the equivariant Dirac operator determined by [Formula: see text] for an equivariant connection A on det W+.