The cohomology of the sporadic groups \(J_2\) and \(J_3\)
Let \(J\) be either the Hall-Janko group \(J_2\) or the Janko group \(J_3\). Let \(S_J\) denote a Sylow \(2\)-subgroup of \(J_2\) (which is isomorphic to a Sylow \(2\)-subgroup of \(J_3\)). For a finite group \(G\) and for a prime \(p\), let \(H^\ast(G;\mathbb{F}_p)\) denote the \(\text{mod}(p)\) cohomology ring of \(G\). The group \(S_J\) contains a single conjugacy class of subgroups \(2^4\) and exactly three conjugacy classes of subgroups \(2\times Q_8\). The cohomology ring \(H^\ast(S_J;\mathbb{F}_2)\) was determined by the second author [in J. Lond. Math. Soc., II. Ser. 51, No. 2, 259-278 (1995; Zbl 0830.20076)]. In this paper, the authors determine \(H^\ast(J;\mathbb{F}_2)\). Their approach is to first use computer calculations in low dimensions to show that \(H^\ast(S_J;\mathbb{F}_2)\) is detected by restriction to a single subgroup \(2^4\) and three copies of a subgroup \(2\times Q_8\) in \(S_J\). Then, using the known normalizer of \(2^4\) in \(J\) and the fusions of the \(2\times Q_8\)'s, they determine \(H^\ast(J;\mathbb{F}_2)\). The reasons for considering these kinds of questions are the following: (1) there is the intrinsic interest in the details of the structure of any of the sporadic groups, (2) there are well-known relations between the structure of the quotient of the cohomology ring \(H^\ast(G;\mathbb{F}_p)\) by the nil-radical, and the structure of the modular representations of \(G\) over the algebraic closure of \(\mathbb{F}_p\); (3) there is a connection with homotopy theory. It is discussed also the relation between this calculation for \(H^\ast(J_3;\mathbb{F}_2)\) and a two-dimensional sporadic geometry for \(J_3\) constructed by \textit{A. J. E. Ryba, S. D. Smith}, and \textit{S. Yoshiara} [in J. Algebra 129, No. 2, 279-311 (1990; Zbl 0693.20015)]. Moreover, it is given a description of the Quillen complex \(\Delta\) for \(J_2\), with quotient \(\Delta/J_2\) which is a three-dimensional complex with 9 vertices, 23 edges, 25 faces, and 10 quads. The quotient is contractible, being a cone on a two-dimensional subcomplex.
sporadic Janko groups \(J_2\) and \(J_3\)
scientific article; zbMATH DE number 1275259