complete growth series of coxeter groups with more than three generators

Complete Growth Series of Coxeter Groups with more than Three Generators

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 and products of groups

The property of having a rational complete growth series is shown to be preserved by direct and graph products, as well as certain free products with amalgamation. These results extend earlier work of Alonso, Chiswell and Lewin.

groups of order 2048 with three generators and three relations

it is shown that there are exactly seventy-eight 3-generator 2-groups of order $2^{11}$ with trivial schur multiplier. we then give 3-generator, 3-relation presentations for forty-eight of them proving that these groups have deficiency zero.

Automorphisms of Coxeter Groups of Rank Three

If W is an infinite rank 3 Coxeter group, whose Coxeter diagram has no infinite bonds, then the automorphism group of W is generated by the inner automorphisms and any automorphisms induced from automorphisms of the Coxeter diagram. Indeed Aut(W ) is the semi-direct product of Inn(W ) and the group of graph automorphisms.

Essays on Coxeter groups Coxeter elements in finite Coxeter groups

A finite Coxeter group possesses a distinguished conjugacy class of Coxeter elements. The literature about these is very large, but it seems to me that there is still room for a better motivated account than what exists. The standard references on thismaterial are [Bourbaki:1968] and [Humphreys:1990], butmy treatment follows [Steinberg:1959] and [Steinberg:1985], from which the clever parts of ...