We classify the maximal irreducible periodic subgroups of PGL(q, $$𝔽$$ ), where $$𝔽$$ is a field of positive characteristic p transcendental over its prime subfield, q = p is prime, and $$𝔽$$ × has an element of order q. That is, we construct a list of irreducible subgroups G of GL(q, $$𝔽$$ ) containing the centre $$𝔽$$ ×1q of GL(q, $$𝔽$$ ), such that G/$$𝔽$$ ×1q is a maximal periodic subgroup of PGL(q, $$𝔽$$ ), and if H is another group of this kind then H is GL(q, $$𝔽$$ )-conjugate to a group in the list. We give criteria for determining...

We describe the finite groups satisfying one of the following conditions: all maximal subgroups permute with all subnormal subgroups, (2) all maximal subgroups and all Sylow $p$-subgroups for $p<7$ permute with all subnormal subgroups.

In this paper, we give a generalization of Baer Theorem on the injective property of divisible abelian groups. As consequences of the obtained result we find a sufficient condition for a group $G$ to express as semi-direct product of a divisible subgroup $D$ and some subgroup $H$. We also apply the main Theorem to the $p$-groups with center of index $p^2$, for some prime $p$. For these groups we compute $N_c\left(G\right)$ the number of conjugacy classes and $N_a$ the number of abelian maximal subgroups and $N_{na}$ the number of nonabelian...