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In this paper, we characterize and compute the mixed and non-mixed basis of Dihedral groups. Also, by computing the conjugacy classes, we describe all the mixed and non-mixed normal subgroups of Dihedral Groups.

The Chevalley–Dickson simple group G24 of Lie type G2 over the Galois field GF4 and of order 251596800=212.33.52.7.13 has a class of maximal subgroups of the form 24+6:A5×3, where 24+6 is a special 2-group with center Z24+6=24. Since 24 is normal in 24+6:A5×3, the group 24+6:A5×3 can be constructed as a nonsplit extension group of the form G¯=24·26:A5×3. Two inertia factor groups, H1=26:A5×3 and H2=26:6×2, are obtained if G¯ acts on 24. In this paper, the author presents a method to compute all projective character tables of H2. These tables become very useful if one wants to construct the ordinary character table of G¯ by means of Fischer–Clifford theory. The method presented here is very effective to compute the irreducible projective character tables of a finite soluble group of manageable size.

Assume p is an odd prime. We investigate finite p-groups all of whose minimal nonabelian subgroups are of order p3. Let P1-groups denote the p-groups all of whose minimal nonabelian subgroups are nonmetacyclic of order p3. In this paper, the P1-groups are classified, and as a by-product, we prove the Hughes’ conjecture is true for the P1-groups.

Let $\\unicode[STIX]{x1D70E}=\\{\\unicode[STIX]{x1D70E}_{i}\\mid i\\in I\\}$ be a partition of the set of all primes $\\mathbb{P}$ . Let $\\unicode[STIX]{x1D70E}_{0}\\in \\unicode[STIX]{x1D6F1}\\subseteq \\unicode[STIX]{x1D70E}$ and let $\\mathfrak{I}$ be a class of finite $\\unicode[STIX]{x1D70E}_{0}$ -groups which is closed under extensions, epimorphic images and subgroups. We say that a finite group $G$ is $\\unicode[STIX]{x1D6F1}_{\\mathfrak{I}}$ - primary provided $G$ is either an $\\mathfrak{I}$ -group or a $\\unicode[STIX]{x1D70E}_{i}$ -group for some $\\unicode[STIX]{x1D70E}_{i}\\in \\unicode[STIX]{x1D6F1}\\setminus \\{\\unicode[STIX]{x1D70E}_{0}\\}$ and we say that a subgroup $A$ of an arbitrary group $G^{\\ast }$ is $\\unicode[STIX]{x1D6F1}_{\\mathfrak{I}}$ - subnormal in $G^{\\ast }$ if there is a subgroup chain $A=A_{0}\\leq A_{1}\\leq \\cdots \\leq A_{t}=G^{\\ast }$ such that either $A_{i-1}\\unlhd A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $\\unicode[STIX]{x1D6F1}_{\\mathfrak{I}}$ -primary for all $i=1,\\ldots ,t$ . We prove that the set ${\\mathcal{L}}_{\\unicode[STIX]{x1D6F1}_{\\mathfrak{I}}}(G)$ of all $\\unicode[STIX]{x1D6F1}_{\\mathfrak{I}}$ -subnormal subgroups of $G$ forms a sublattice of the lattice of all subgroups of $G$ and we describe the conditions under which the lattice ${\\mathcal{L}}_{\\unicode[STIX]{x1D6F1}_{\\mathfrak{I}}}(G)$ is modular.

In topological dynamics, the Of course, the statement is preceded by the presentation of the concepts of an entropy structure and its superenvelopes, adapted from the case of

Let and be subgroups in a finite group  . Then is (hereditarily) -permutable with  if for some (for some ). A subgroup in is (hereditarily) -permutable in if is (hereditarily) -permutable with all subgroups in  . The article deals with the structure of  such that the normalizers of Sylow subgroups are (hereditarily) -permutable.

This book classifies the maximal subgroups of the almost simple finite classical groups in dimension up to 12; it also describes the maximal subgroups of the almost simple finite exceptional groups with socle one of Sz(q), G2(q), 2G2(q) or 3D4(q). Theoretical and computational tools are used throughout, with downloadable Magma code provided. The exposition contains a wealth of information on the structure and action of the geometric subgroups of classical groups, but the reader will also encounter methods for analysing the structure and maximality of almost simple subgroups of almost simple groups. Additionally, this book contains detailed information on using Magma to calculate with representations over number fields and finite fields. Featured within are previously unseen results and over 80 tables describing the maximal subgroups, making this volume an essential reference for researchers. It also functions as a graduate-level textbook on finite simple groups, computational group theory and representation theory.

For a group G and $m\\ge 1$ , let $G^m$ denote the subgroup generated by the elements $g^m$ , where g runs through G. The subgroups not of the form $G^m$ are the nonpower subgroups of G. We classify the groups with at most nine nonpower subgroups.

In the 2006 edition of the Kourovka Notebook , Berkovich poses the following problem (Problem 16.13): Let   $p$   be a prime and   $P$   be a finite   $p$ - group . Can   $P$   have every maximal subgroup special? We show that the structure of such groups is very restricted, but for all primes there are groups of arbitrarily large size with this property.