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 7→ [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.

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[1] Blyth RD, Fumagalli F, Morigi M. Some structural results on the non-abelian tensor square of groups. J Group Theory 2010; 13: 8394.
[2] Brown R, Johnson DL, Robertson EF. Some computations of nonabelian tensor products of groups. J Algebra 1987; 111: 177202.
[3] Brown R, Loday JL. Van Kampen theorems for diagrams of spaces. Topology 1987; 26: 311335.
[4] Brown R, Higgins PJ, Sivera R. Nonabelian Algebraic Topology. Z¨urich, Switzerland: EMS Publishing House, 2011.
[5] Ellis G. On the tensor square of a prime power group. Arch Math (Basel) 1996: 66: 467469.
[6] Ellis G. On the relation between upper central quotients and lower central series of a group. Trans Amer Math Soc 2001; 353: 42194234.
[7] Ellis G, McDermott A. Tensor products of prime power groups. J Pure Appl Algebra 1998; 132: 119128.
[8] Hatcher A. Algebraic Topology. Cambridge, UK: Cambridge University Press, 2002.
[9] Jafari SH, Moghaddam MRR, Niroomand P. Some properties of the tensor centre of groups. J Korean Math Soc 2009; 46: 249256.
[10] Jafari SH. A bound on the order of nonabelian tensor square of a prime power group. Comm Algebra 2012; 40: 528530.
[11] Karpilovsky G. The Schur Multiplier. Oxford, UK: Clarendon Press, 1987.
[12] Moghaddam MRR, Niroomand P. Some properties of certain subgroups of tensor squares of p-groups. Comm Algebra 2012; 40: 11881193.
[13] Niroomand P, Russo FG. A note on the exterior centralizer. Arch Math (Basel) 2009; 93: 505512.
[14] Niroomand P, Russo FG. An improvement of a bound of Green. J Algebra Appl 2012; 11: 1250116.
[15] Rocco NR. On a construction related to the nonabelian tensor square of a group. Bol Soc Brasil Mat 1991; 22: 6379.
[16] Rocco NR. A presentation for crossed embedding of finite solvable groups. Comm Algebra 1994; 22: 19751998.
[17] Rotman J. An Introduction to Algebraic Topology. Berlin, Germany: Springer, 1988.