‎In this paper we study codes $C_p(overline{{rm HiS}})$ where $p =3,7‎, ‎11$ defined by the 3‎- ‎7‎- ‎and 11-modular representations of the simple sporadic group ${rm HS}$ of Higman and Sims of degree 100‎. ‎With exception of $p=11$ the codes are those defined by the row span of the adjacency matrix of the complement of the Higman-Sims graph over $GF(3)$ and $GF(7).$ We show that these codes ha...

3E. L. Post, J. Symb. Logic, 12, 1-11, 1947. 4In the sense of Post, the word problem for Z is reducible to that for sp (2:). 5 W. Magnus, Math. Annalen, 106, 295-307, 1932. 6 DanaScott, abstract, J. Symb. Logic, 21, 111-112, 1956. 7Marshall Hall, J. Symb. Logic, 14, 115-118, 1949. 8 G. Higman, B. H. Neumann, and H. Neumann, J. London Math. Soc., 24, 247-254, 1949. 9 As defined, UG, Part I, p. 2...

We consider the subgroup lpGk,1 of length preserving elements of the Thompson-Higman group Gk,1 and we show that all elements of Gk,1 have a unique lpGk,1 · Fk,1 factorization. This applies to the Thompson-Higman group Tk,1 as well. We show that lpGk,1 is a “diagonal” direct limit of finite symmetric groups, and that lpTk,1 is a k ∞ Prüfer group. We find an infinite generating set of lpGk,1 whi...

We consider the wreath product of two symmetric groups as a group blocks permutations and we study its conjugacy classes. give polynomiality property for structure coefficients center algebras. This allows us to recover an old result Farahat Higman about algebra generalize our recent hyperoctahedral algebra.

We prove that the number of groups of order p whose Frattini subgroup is central is for fixed n a PORC (‘polynomial on residue classes’) function of p. This extends a result of G. Higman.

In this paper, we will prove the inevitable non-uniformity of two constructions from combinatorial group theory related to the word problem for finitely generated groups and the Higman-Neumann-Neumann Embedding Theorem.