Finite groups that need more generators than any proper quotient

Finite Groups That Need More Generators than Any Proper Quotient

A structure theorem is proved for finite groups with the property that, for some integer m with m 1⁄2 2, every proper quotient group can be generated by m elements but the group itself cannot. 1991 Mathematics subject classification (Amer. Math. Soc.): 20D20.

Complete Growth Series of Coxeter Groups with more than Three Generators

Groups without Proper Isomorphic Quotient Groups

If ƒ is a homomorphism of the group G, and if g is an isomorphism of the image group G, then f g is a homomorphism of G too ; and this homomorphism is an isomorphism if, and only if, ƒ is an isomorphism. Consequently the following three properties of the group G imply each other. (1) Homomorphisms of G upon isomorphic groups are isomorphisms. (2) Homomorphisms of G upon itself are isomorphisms....

Finite groups have even more centralizers

For a finite group $G$‎, ‎let $Cent(G)$ denote the set of centralizers of single elements of $G$‎. ‎In this note we prove that if $|G|leq frac{3}{2}|Cent(G)|$ and $G$ is 2-nilpotent‎, ‎then $Gcong S_3‎, ‎D_{10}$ or $S_3times S_3$‎. ‎This result gives a partial and positive answer to a conjecture raised by A‎. ‎R‎. ‎Ashrafi [On finite groups with a given number of centralizers‎, ‎Algebra‎ ‎Collo...

complete growth series of coxeter groups with more than three generators