In the mathematics of

graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

and finite groups, a prime graph is an

undirected graph In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' v ...

defined from a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

. These graphs were introduced in a 1981 paper by J. S. Williams, credited to unpublished work from 1975 by K. W. Gruenberg and O. Kegel.

Definition

The prime graph of a group has a

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

for each prime number that divides the

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...

(number of elements) of the given group, and an

edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed b ...

connecting each pair of prime numbers

p

and

q

for which there exists a group element with order

pq

. Equivalently, there is an edge from

p

q

whenever the given group contains commuting elements of order

p

and of order

q

, or whenever the given group contains a

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...

of order

pq

as one of its subgroups.

Properties

Certain

finite simple group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

s can be recognized by the degrees of the vertices in their prime graphs. The connected components of a prime graph have

diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...

at most five, and at most three for

solvable group In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group (mathematics), group that can be constructed from abelian groups using Group extension, extensions. Equivalently, a solvable group is a ...

s. When a prime graph is a

tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...

, it has at most eight vertices, and at most four for solvable groups.

Related graphs

Variations of prime graphs that replace the existence of a cyclic subgroup of order

pq

, in the definition for adjacency in a prime graph, by the existence of a subgroup of another type, have also been studied. Similar results have also been obtained from a related family of graphs, obtained from a finite group through the degrees of its

characters Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...

rather than through the orders of its elements.

References

{{reflist, refs= {{citation , last1 = Abe , first1 = Seiichi , last2 = Iiyori , first2 = Nobuo , doi = 10.14492/hokmj/1350912979 , issue = 2 , journal = Hokkaido Mathematical Journal , mr = 1776716 , pages = 391–407 , title = A generalization of prime graphs of finite groups , volume = 29 , year = 2000, url = http://projecteuclid.org/euclid.hokmj/1350912979 {{citation , last = Tong-Viet , first = Hung P. , doi = 10.1016/j.jalgebra.2012.12.024 , journal = Journal of Algebra , mr = 3017021 , pages = 196–206 , title = Groups whose prime graphs have no triangles , volume = 378 , year = 2013, s2cid = 119118934 , arxiv = 1303.3457 {{citation , last1 = Moghaddamfar , first1 = A. R. , last2 = Zokayi , first2 = A. R. , last3 = Darafsheh , first3 = M. R. , doi = 10.1142/S1005386705000398 , issue = 3 , journal = Algebra Colloquium , mr = 2144997 , pages = 431–442 , title = A characterization of finite simple groups by the degrees of vertices of their prime graphs , volume = 12 , year = 2005 {{citation , last = Lucido , first = Maria Silvia , author-link = Maria Silvia Lucido , doi = 10.1515/jgth.1999.011 , issue = 2 , journal = Journal of Group Theory , mr = 1681526 , pages = 157–172 , title = The diameter of the prime graph of a finite group , volume = 2 , year = 1999 , zbl = 0921.20020 {{citation , last = Lucido , first = Maria Silvia , author-link = Maria Silvia Lucido , issue = 1 , journal = Bollettino della Unione Matematica Italiana , mr = 1881928 , pages = 131–148 , title = Groups in which the prime graph is a tree , volume = 5 , year = 2002 , zbl = 1097.20022 {{citation , last = Williams , first = J. S. , doi = 10.1016/0021-8693(81)90218-0 , issue = 2 , journal = Journal of Algebra , mr = 617092 , pages = 487–513 , title = Prime graph components of finite groups , volume = 69 , year = 1981, doi-access = free Application-specific graphs Finite groups