Let C be the centralizer in a finite Weyl group of an elementary abelian 2-subgroup. We show that every complex representation of C can be realized over the field of rational numbers. The same holds for a Sylow 2-subgroup of C.

A subgroup H is said to be nc-supplemented in a group G if there is a subgroup K ≤ G such that HK G and H ∩ K is contained in HG, the core of H in G. We characterize the solvability of finite groups G with some subgroups of Sylow subgroups nc-supplemented in G. We also give a result on c-supplemented subgroups.

The totality formed by all the operators of any group (G) which are common to all the invariant subgroups of prime index (p) constitutes a characteristic subgroup, and the corresponding quotient group is the abelian group of order pK and of type (1, 1, 1, ■■■)-\ The number of the invariant subgroups of index p is therefore pK — 1/p — 1. The given totality includes all the operators of G which a...

We show that the homotopy type of a finite oriented Poincar\'{e} 4-complex is determined by its quadratic 2-type provided fundamental group and has dihedral Sylow 2-subgroup. By combining with results Hambleton-Kreck Bauer, this applies in case smooth 4-manifolds whose subgroup SO(3). An important class examples are elliptic surfaces group.

A nilpotent group G is a finite group that is the direct product of its Sylow p-subgroups. Theorem 1.1 (Fitting's Theorem) Let G be a finite group, and let H and K be two nilpotent normal subgroups of G. Then HK is nilpotent. Hence in any finite group there is a unique maximal normal nilpotent subgroup, and every nilpotent normal subgroup lies inside this; it is called the Fitting subgroup, and...

The novel notion of rigid commutators is introduced to determine the sequence logarithms indices a certain normalizer chain in Sylow 2-subgroup symmetric group on 2^n letters. terms this are proved be those partial sums partitions an integer into at least two distinct parts, that relates famous Euler's partition theorem.

The totality formed by all the operators of any group (G) which are common to all the invariant subgroups of prime index (p) constitutes a characteristic subgroup, and the corresponding quotient group is the abelian group of order pK and of type (1, 1, 1, ■■■)-\ The number of the invariant subgroups of index p is therefore pK — 1/p — 1. The given totality includes all the operators of G which a...

We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then Sylow p-subgroup has order 5, 7 or is isomorphic to one non-abelian 2-groups 8.

Let G be a finite group, and let F be a formation of finite group. We say that a subgroup H of G is p F -normal in G if there exists a normal subgroup T of G such that HT is a permutable Hall subgroup of G and G G H H T H / ) ( is contained in the F-hypercenter ) / ( G F H G Z of G H G / . In this note, we get some results about the p F -normal subgroups and then use them to study the structure...

We consider the problem of testing isomorphism of groups of order n given by Cayley tables. The trivial nlogn bound on the time complexity for the general case has not been improved over the past four decades. Recently, Babai et al. (following Babai et al. in SODA 2011) presented a polynomial-time algorithm for groups without abelian normal subgroups, which suggests solvable groups as the hard ...