A note on coset complexes of $p$-subgroups

Abstract

This paper investigates the coset complexes of $p$-subgroups in finite groups. Given a finite group $G$ and a prime $p$, we define \( \mathscr{C}_p(G) \) as the poset of all cosets of $p$-subgroups of $G$. We construct a probability function \( P_p(G,s) \) with group-theoretic connections, strengthen the congruence formula of the $p$-local Euler characteristic of \( \mathscr{C}_p(G) \), and analyze the connectivity of \( \mathscr{C}_p(G) \).

Keywords

• coset complexes
• probability function
• Möbius function
• connectivity

Supporting Institution

the Natural Science Foundation of Shanghai，the National Natural Science Foundation of China

Project Number

24ZR1422800，12471018，12171302

Thanks

The second author is supported by the Natural Science Foundation of Shanghai (24ZR1422800) and the National Natural Science Foundation of China (12471018); The third author is supported by the National Natural Science Foundation of China (12171302).

References

1. [1] E. Detomi and A. Lucchini, Recognizing soluble groups from their probabilistic zeta functions, Bull. London Math. Soc. 35 (5), 659–664, 2003.
2. [2] E. Detomi and A. Lucchini,Some generalizations of the probabilistic zeta function, Ischia Group Theory 2006, 56–72, 2007.
3. [3] H. Gu, H. Meng, and X. Guo, Coset complexes of p-subgroups in finite groups, Reprint, arXiv:2503.06379, 2025.
4. [4] P. Hall. The Eulerian functions of a finite group, Q. J. Math. 7 (1), 134–151, 1936.
5. [5] C. Y. Ho, Finite groups in which two different Sylow p-subgroups have trivial intersection for an odd prime p, J. Math. Soc. Japan, 31 (4), 669–675, 1979.
6. [6] D. Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group, Advances in Math. 28 (2), 101–128, 1978.
7. [7] E. Snapper, Counting p-subgroups Proc. Amer. Math. Soc. 39 (1), 81–82, 1973.
8. [8] M. Suzuki, Finite groups of even order in which Sylow 2-groups are independent, Ann. of Math. 80 (1), 58–77, 1964.

9. [9] M.Wachs, Poset topology: tools and applications, Reprint, arXiv: math/0602226, 2006.