A finite non-abelian group G is called metahamiltonian if every subgroup of either abelian or normal in G. If non-nilpotent, then the structure has been determined. nilpotent, determined by its Sylow subgroups. However, classification p-groups an unsolved problem. In this paper, are completely classified up to isomorphism.

We show that the character table of a finite group $G$ determines whether Sylow 2-subgroup is generated by 2 elements, in terms Galois action on characters. Our proof this result requires use Classification Finite Simple Groups and provides new evidence for so-far elusive Alperinâ??McKayâ??Navarro conjecture.

Abstract A rigid automorphism of a linking system is an that restricts to the identity on Sylow subgroup. inner conjugation by element in center At odd primes, it known each centric inner. We prove group outer automorphisms at prime $2$ elementary abelian and splits over subgroup automorphisms. In second result, we show if finite G for , then $p'$ -order modulo automorphisms, provided has no no...

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. Towards this end, the development of the theory of groups of finite Morley rank has achieved a good theory of Sylow 2-subgroups. It is now common practice to divide the Cherlin-Zilber conjecture into different cases depending on the nature of the...

Let G $G$ be a finite group admitting coprime automorphism α $\alpha$ . J ( ) $J_G(\alpha )$ denote the set of all commutators [ x , ] $[x,\alpha ]$ where $x$ belongs to an -invariant Sylow subgroup We show that $[G,\alpha is soluble or nilpotent if and only any generated by pair elements orders from nilpotent, respectively.

As a counterpart for the prime 2 to Glauberman’s ZJ-theorem, Stellmacher proves that any nontrivial 2-group S has a nontrivial characteristic subgroup W (S) with the following property. For any finite Σ4-free group G, with S a Sylow 2-subgroup of G and with O2(G) self-centralizing, the subgroup W (S) is normal in G. We generalize Stellmacher’s result to fusion systems. A similar construction of...

A perfect code in a graph $$\Gamma $$ is subset C of $$V(\Gamma )$$ such that no two vertices are adjacent and every vertex ){\setminus } C$$ to exactly one C. Let G be finite group G. Then said if there exists Cayley admiting as code. It proved subgroup H only Sylow 2-subgroup This result provides way simplify the study codes general groups 2-groups. As an application, criterion for determinin...

Let G be a permutation group of set ? and k positive integer. The k-closure is the greatest (w.r.t. inclusion) subgroup G(k) in Sym(?) which has same orbits as under componentwise action on ?k. It proved that finite nilpotent coincides with direct product k-closures all its Sylow subgroups.