Pairs of groups having a common 2-subgroup of prime indices (Q1067000)

This paper deals with the problem of investigating the pairs (G,H) of finite groups satisfying the conditions: (a) G and H have a common 2- subgroup S. (b) Both the indices \(q=[G:S]\) and \(r=[H:S]\) are odd prime integers. (c) No nonidentity subgroup of S is normal both in G and in H. \textit{D. M. Goldschmidt} [in Ann. Math., II. Ser. 111, 377-406 (1980; Zbl 0475.05043)] considered the special case \(q=r=3\) and obtained a complete description of the possible pairs (G,H). Then \textit{P. Rowley} [in Math. Z. 181, 293-312 (1982; Zbl 0492.20014)] investigated up to some extent the general case under the additional condition (d) the centralizer of the maximal normal 2-subgroup \(O_ 2(G)\) of G in G is contained in itself and the centralizer of the maximal normal 2-subgroup \(O_ 2(H)\) of H in H is contained in itself and showed that if furthermore \([S:O_ 2(G)]=2=[S:O_ 2(H)]\) then \(q=r=3\). The present author analyses the pairs (G,H) of finite groups satisfying the conditions (a), (b), (c), (d) by defining four types of such pairs and proving several theorems whose statements involve these four types.