mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Higman group, introduced by , was the first example of an infinite

finitely presented group In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...

with no non-trivial finite quotients. The quotient by the maximal proper

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...

is a finitely generated infinite

simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...

. later found some finitely presented infinite groups that are simple if is even and have a simple subgroup of index 2 if is odd, one of which is one of the

Thompson groups In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted F \subseteq T \subseteq V, that were introduced by Richard Thompson in some unpublished handwritten notes i ...

. Higman's group is generated by 4 elements with the relations :

a^ba=b^2,\quad b^cb=c^2,\quad c^dc=d^2,\quad d^ad=a^2.

References

* *{{Citation , last1=Higman , first1=Graham , author1-link=Graham Higman , title=Finitely presented infinite simple groups , url=https://books.google.com/books?id=LPvuAAAAMAAJ , publisher=Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra , series=Notes on Pure Mathematics , isbn=978-0-7081-0300-5 , mr=0376874 , year=1974 , volume=8 Group theory