\(M_ p\)-groups
Let G be a group, B be an infinite normal divisible abelian p-subgroup satisfying the minimal condition, a be an element of order p, and assume that the following conditions hold: a) the locally finite p-subgroups of \(C_ G(a)B/B\) are finite, b) if some divisible abelian p-subgroup C of G is contained in the set \(\cup_{g\in G}gp(a,a^ g)\), then \(C\leq B\). Such a group G is said to be an \(M_ p\)-group with kernel B and handle (a). The following fundamental theorem is proved. Let G be a group without involutions, and let B be an infinite divisible abelian p- subgroup, satisfying the conditions: 1) \(H=N_ G(B)\) is an \(M_ p\)- group with kernel B and handle (a) such that the locally finite p- subgroups in \(C_ G(a)\) are finite, 2) \(gp(a,a^ g)\) is finite for every \(g\in G\setminus H\), 3) \(| C_ G(a):H\cap C_ G(a)|\) is finite and \(H\cap C_ G(a)\) contains all the p'-elements of finite order from \(C_ G(a)\), 4) if Q is a finite (a)-invariant q-subgroup from H satisfying \(Q\cap C_ G(a)\neq 1\) and \(q\neq p\), then \(N_ G(Q)\leq H\). Then B is normal in G. Some examples of groups are given in which the condition 4) is automatically satisfied, while each of the conditions 1)- 3) is independent of the remaining ones.
infinite normal divisible abelian p-subgroup
scientific article; zbMATH DE number 3902891