Turkish Journal of Mathematics
DOI
10.3906/mat-1302-50
Abstract
The nonabelian tensor square G \otimes G of a group G of G = p^n and G' = p^m (p prime and n,m \ge 1) satisfies a classic bound of the form G \otimes G \le 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 \otimes y \in G \otimes G \mapsto [x,y] \in G'. We prove that G \otimes G \le p^{(n-1)(n-m)+2}, sharpening not only G \otimes G \le 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.
Keywords
Schur multipliers, p--groups, nonabelian tensor square, homotopy
First Page
664
Last Page
671
Recommended Citation
NIROOMAND, PEYMAN and RUSSO, FRANCESCO
(2014)
"On the size of the third homotopy group of the suspension of an Eilenberg--MacLane space,"
Turkish Journal of Mathematics: Vol. 38:
No.
4, Article 6.
https://doi.org/10.3906/mat-1302-50
Available at:
https://journals.tubitak.gov.tr/math/vol38/iss4/6
