On the size of the third homotopy group of the suspension of an Eilenberg-MacLane space

The nonabelian tensor square G ⊗ G of a group G of |G| = p^n and |G′| = p^m ( p prime and n, m ≥ 1) satisfies a classic bound of the form |G ⊗ G| ≤ p^n(n−m). This allows us to give an upper bound for the order of the third homotopy group π3(SK(G, 1)) of the suspension of an Eilenberg–MacLane space K(G, 1), because π3(K(G, 1)) is isomorphic to the kernel of κ : x ⊗ y ∈ G ⊗ G t→ [x, y] ∈ G′. We prove that |G ⊗ G| ≤ p^(n−1)(n−m)+2, sharpening not only |G ⊗ G| ≤ p^n(n−m) but also supporting a recent result of Jafari on the topic. Consequently, we discuss restrictions on the size of π3(SK(G, 1)) based on this new estimation.