A block-theory-free characterization of
The order of every finite group can be expressed as a product of coprime positive integers such that is a connected component of the prime graph of . The integers are called the order components of . Some non-abelian simple groups are known to be uniquely determined by their order components. As the main result of this paper, we show that the projective symplectic groups where are also uniquely determined by their order components. As corollaries of this result, the validities of a...
We proved that the symplectic groups , where is a Fermat prime number is uniquely determined by its order, the first largest element orders and the second largest element orders.
Let be a finite group and be the set of element orders of . Let and be the number of elements of order in . Set . In fact is the set of sizes of elements with the same order in . In this paper, by and order, we give a new characterization of finite projective special linear groups over a field with elements, where is prime. We prove the following theorem: If is a group such that and consists of , , and some numbers divisible by , where is a prime greater than...
Let denote the set of element orders of a finite group . If is a finite non-abelian simple group and implies contains a unique non-abelian composition factor isomorphic to , then is called quasirecognizable by the set of its element orders. In this paper we will prove that the group is quasirecognizable.
We investigate loops defined upon the product by the formula , where , for appropriate parameters . Each such loop is coupled to a 2-cocycle (in the group-theoretical sense) and this connection makes it possible to prove that the loop possesses a metacyclic inner mapping group. If , then the loop is an A-loop. Questions of isotopism and isomorphism are considered in detail.
We report on a partial solution of the conjecture that the class of finite solvable groups can be characterised by 2-variable identities. The proof requires pieces from number theory, algebraic geometry, singularity theory and computer algebra. The computations were carried out using the computer algebra system SINGULAR.