On odd prime components of cohomologies of sporadic simple groups and the rings of universal stable elements
Let \(G\) be a finite group and \(p\) be a prime. Let \(H^*(G)\) be the cohomology ring of \(G\) over \(Z\) and let \(H^*(G)_{(p)}=H^*(G)\otimes Z_{(p)}\). Denote by \(E\) the extraspecial \(p\)-group of order \(p^3\) and of exponent \(p\). Given a \(p\)-group \(P\), \textit{L. Evens} and \textit{S. Priddy} [Q. J. Math., Oxf. II. Ser. 40, No. 160, 399-407 (1989; Zbl 0687.20047)] defined the ring of universal stable elements by \(I(P)=\bigcup_G\text{Im(res }H^*(G)\to H^*(P))\), where \(G\) runs over all finite groups with \(P\) as a Sylow \(p\)-subgroup. In this paper, it is proved that if \(G\) has a Sylow \(p\)-subgroup isomorphic to \(E\) for an odd prime \(p\) and \(G\) contains \(\text{Hol}(Z_{(p)}\times Z_{(p)})\) as a subgroup, and all subgroups isomorphic \(Z_{(p)}\times Z_{(p)}\) are conjugated, then \(H^*(G)_{(p)}=I(E)\) (Theorem 0.1). S. Yoshiara gave the authors a list of sporadic simple groups \(G\) with Sylow \(p\)-subgroup isomorphic to \(E\) for odd \(p\), together with almost all information necessary to obtain cohomologies of \(G\). The authors study systematically odd prime components of cohomologies for groups from this list. In particular, it is shown that the Rudvalis group \(Ru\) and the largest Janko group \(J_4\) for \(p=3\), and the Thompson group \(Th\) for \(p=5\) satisfy the assumption of Theorem 0.1, so \(H^*(G)_{(p)}=I(E)\) for these groups and primes. The paper continues the investigations of cohomologies of sporadic simple groups in works of \textit{C. B. Thomas} [Lect. Notes Math. 1474, 371-381 (1991; Zbl 0737.20023)], \textit{D. J. Green} [Math. Proc. Camb. Philos. Soc. 113, No. 2, 253-266 (1993; Zbl 0828.20037)], \textit{C. B. Thomas} [Contemp. Math. 158, 307-318 (1994; Zbl 0846.20058)], \textit{I. Leary} [``The cohomology of certain finite groups'', Thesis, Cambridge Univ. (1990)], \textit{R. J. Milgram} and \textit{M. Tezuka} [Bol. Soc. Mat. Mex., III. Ser. 1, No. 2, 91-108 (1995; see the preceding review Zbl 0855.20044)].
scientific article; zbMATH DE number 923695