A $P_4$-free graph is called a cograph. In this paper we partially characterize finite groups whose power As will see, problem generalization of the determination in which every element has prime order, first raised by Graham Higman 1957 and fully solved very recently. First determine all $G$ $H$ for $G\times H$ We show that cograph can be characterised condition only involving elements orders ...

[4] Lev Glebsky and Luis Manuel Rivera, Sofic groups and profinite topology on free groups, J. Algebra 320 (2008), no. 9, 3512-3518. [2] Misha Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 2, 109-197. [3] Gábor Elek and Endre Szabó, Hyperlinearity, essentially free actions and Linvariants. The sofic property, Math. Ann. 332 (2005), no. 2, 421-441...

William W. Boone and Graham Higman proved that a finitely generated group has soluble word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group. We prove the exact analogue for lattice-ordered groups: Theorem: A finitely generated lattice-ordered group has soluble word problem if and only if it can be `-embedded in an `-simple lattice-or...

It all started with a theorem of Miller [14]: the classical modular group PSL2Z‘ has among its homomorphic images every alternating group, except A6; A7; and A8. In the late 1960s Graham Higman conjectured that any (finitely generated non-elementary) Fuchsian group has among its homomorphic images all but finitely many of the alternating groups. This reduces to an investigation of the cocompac...

We show that for the family of Church-Rosser languages the Higman-Haines sets, which are the sets of all scattered subwords of a given language and the sets of all words that contain some word of a given language as a scattered subword, cannot be effectively constructed, although these both sets are regular for any language. This nicely contrasts the result on the effectiveness of the Higman-Ha...

The Higman-Sims design is an incidence structure of 176 points and 176 blocks of cardinality 50 with every two blocks meeting in 14 points. The automorphism group of this design is the Higman-Sims simple group. We demonstrate that the point set and the block set of the Higman-Sims design can be partitioned into subsets X1, X2, . . . , X11 and B1, B2, . . . , B11, respectively, so that the subst...

A not so well-known result in formal language theory is that the Higman-Haines sets for any language are regular [11, Theorem 4.4]. It is easily seen that these sets cannot be effectively computed in general. The Higman-Haines sets are the languages of all scattered subwords of a given language as well as the sets of all words that contain some word of a given language as a scattered subword. R...

Abstract In the context of Higman embeddings recursive groups into finitely presented groups, we suggest an approach, termed