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

The nonabelian tensor square G circle times G of a group G of vertical bar G vertical bar = p(n) and vertical bar G'vertical bar = p(m) (p prime and n, m >= 1) satisfies a classic bound of the form vertical bar G circle times G vertical bar <= p(n(n-m)). This allows us to give an upper bound for the order of the third homotopy group pi(3)(SK(G, 1)) of the suspension of an Eilenberg MacLane space K(G,1), because pi(3)(K(G, 1)) is isomorphic to the kernel of kappa : x circle times y is an element of G circle times G [x, y] is an element of G'. We prove that vertical bar G circle times G vertical bar <= p((n-1)(n-m)+2), sharpening not only vertical bar G circle times G vertical bar <= p(n(n-m)) but also supporting a recent result of Jafari on the topic. Consequently, we discuss restrictions on the size of pi(3)(SK(G, 1)) based on this new estimation.