Let R be a ring with 1, I be a nilpotent subring of R (there exists a natural number n, such that In = 0), and I be generated by {xj |j ∈ J} as ring. Write U = 1 + I, and it is a nilpotent group with class ≤ n - 1. Let G be the subgroup of U which is generated by {1 + xj|j ∈ J}. The group constructed in this paper indicates that the nilpotency class of G can be less than that of U.
LIU HeGuo1, ZHANG JiPing2 & LIAO Jun2 1Department of Mathematics, Hubei University, Wuhan 430062, China
A (δ, g)-cage is a δ-regular graph with girth g and with the least possible number of vertices. In this paper, we show that all (δ, g)-cages with odd girth g ≥ 9 are r-connected, where (r - 1)2 ≤δ + x/δ - 2 〈 r2 and all (δ, g)-cages with even girth g 〉 10 are r-connected, where r is the largest integer satisfying r(r-1)2/4 + 1 + 2r(r - 1) ≤δ. These results support a conjecture of Fkl, Huang and Rodger that all (δ, g)-cages are 6-connected.