Amalgams which involve sporadic simple groups. I
[For part II see the following review Zbl 0885.20014.] Let \(\mathcal S\) and \(\mathcal L\) denote the following sets of groups: \(\mathcal S=\{L\mid F^\ast(L)/Z(F^\ast(L))\) is a sporadic simple group and \(Z(F^*(L))\) has odd order\} and \(\mathcal L=\{L_2(2^n)\), \(U_3(2^n)\), \(SU_3(2^n)\), \(Sz(2^{2n+1})\), \(D_{2p}\), \(S_5\mid n\in\mathbb{N}\), \(p\in\mathbb{P}_0\}\), where \(D_{2p}\) denotes a dihedral group of order \(2p\) and \(\mathbb{P}_0\) is the set of all odd primes. Finally, let \(\Gamma\) be the coset graph associated with a rank 2 amalgam for a group \(G=\langle P_1,P_2\rangle\) over a finite group \(B=P_1\cap P_2\) with \(O^{2'}(P_1)/O_2(O^{2'}(P_1))\in{\mathcal L}\) and \(O^{2'}(P_2)/O_2(O^{2'}(P_2))\in{\mathcal S}\), such that \(\text{Syl}_2(B)\subseteq\text{Syl}_2(P_i)\), \(P_i=O^{2'}(P_i)B\), \(C_{P_i}(O_2(P_i))\leq O_2(P_i)\) (\(i=1,2\)) and \(\text{core}_GB=1\). The authors determine the chief series structure for all possible pairs \((P_1,P_2)\). In particular it follows that \(P_1/O_2(P_1)\cong L_2(2)\) or \(S_5\) and \(P_2/O_2(P_2)\) is isomorphic to one of the groups in \(\{M_{22}\), \(\Aut(M_{22})\), \(3\Aut(M_{22})\), \(M_{23}\), \(M_{24}\), \(Co_2\), \(Co_1\}\). In part I the authors investigate the noncommuting case. They show that \(b=2\) (where \(b\) denotes the minimum distance between pairs of vertices \(\alpha,\alpha'\in\Gamma\), satisfying \(Z_\alpha=\langle\Omega_1(Z(S))\mid S\in\text{Syl}_2(G_\alpha)\rangle\nleq O_2(G_{\alpha'})\)) and that there are three possible pairs \((P_1,P_2)\) (realized in \(J_4\), \(B\) and \(M\)).
scientific article; zbMATH DE number 1051572
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