Embeddings into k-Efficient Groups

We prove a theorem with the following corollary: For each integer k ≥ 1, an arbitrary finite group G embeds into some finite group Gk for which there exists an Eilenberg–Mac Lane CW-space X = K(Gk, 1) whose finite n-skeleton Xn has Euler–Poincaré characteristic χ(Xn) = 1 + (− 1)ndHn(Gk) for all n ≤ k. The theorem can be viewed as a generalisation of a result of J. Harlander [1996, J. Algebra182, 511–521] on the embedding of finite groups into groups with “efficient” presentations.