abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, the group isomorphism problem is the

decision problem In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natura ...

of determining whether two given finite group presentations refer to

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

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 iden ...

s. The isomorphism problem was formulated by

Max Dehn Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Dehn's early life and career took place in Germany. However, he was forced to retire in 1 ...

, and together with the word problem and

conjugacy problem In abstract algebra, the conjugacy problem for a group ''G'' with a given presentation is the decision problem of determining, given two words ''x'' and ''y'' in ''G'', whether or not they represent conjugate elements of ''G''. That is, the probl ...

, is one of three fundamental decision problems in group theory he identified in 1911. All three problems, formulated as ranging over all finitely presented groups, are undecidable. In the case of the isomorphism problem, this means that there does not exist a computer algorithm that takes two finite group presentations and decides whether or not the groups are isomorphic, regardless of how (finitely) much time is allowed for the algorithm to run and how (finitely) much memory is available. In fact the problem of deciding whether a finitely presented group is trivial is undecidable, a consequence of the

Adian–Rabin theorem In the mathematical subject of group theory, the Adyan–Rabin theorem is a result that states that most "reasonable" properties of finitely presentable groups are algorithmically undecidable. The theorem is due to Sergei Adyan (1955) and, indepe ...

due to

Sergei Adian Sergei Ivanovich Adian, also Adyan (; ; 1 January 1931 – 5 May 2020), 4381, and hence for all multiples of those odd integers as well. The solution of the Burnside problem was certainly one of the most outstanding and deep mathematical results ...

and

Michael O. Rabin Michael Oser Rabin (; born September 1, 1931) is an Israeli mathematician, computer scientist, and recipient of the Turing Award. Biography Early life and education Rabin was born in 1931 in Breslau, Germany (today Wrocław, in Poland), th ...

. However, there are some classes of finitely presented groups for which the restriction of the isomorphism problem is known to be decidable. They include finitely generated abelian groups,

finite groups In abstract algebra, a finite group is a group (mathematics), group whose underlying set is finite set, finite. Finite groups often arise when considering symmetry of Symmetry in mathematics, mathematical or Symmetry (physics), physical objects, ...

, Gromov-hyperbolic groups,

virtually In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group ''G'' is said to b ...

torsion-free relatively hyperbolic groups with

nilpotent In mathematics, an element x of a ring (mathematics), ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term, along with its sister Idempotent (ring theory), idem ...

parabolics, one-relator groups with non-trivial

center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...

, and two-generator one-relator groups with torsion. The group isomorphism problem, restricted to the groups that are given by multiplication tables, can be reduced to a

graph isomorphism problem The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. The problem is not known to be solvable in polynomial time nor to be NP-complete, and therefore may be in the computational c ...

but not vice versa. Both have quasi-polynomial-time algorithms, the former since 1978 attributed to

Robert Tarjan Robert Endre Tarjan (born April 30, 1948) is an American computer scientist and mathematician. He is the discoverer of several graph theory algorithms, including his strongly connected components algorithm, and co-inventor of both splay trees a ...

and the latter since 2015 by

László Babai László "Laci" Babai (born July 20, 1950, in Budapest) a fellow of the American Academy of Arts and Sciences, and won the Knuth Prize. Babai was an invited speaker at the International Congresses of Mathematicians in Kyoto (1990), Zürich (199 ...

. A small but important improvement for the case

p-group In mathematics, specifically group theory, given a prime number ''p'', a ''p''-group is a group in which the order of every element is a power of ''p''. That is, for each element ''g'' of a ''p''-group ''G'', there exists a nonnegative integ ...

s of class 2 was obtained in 2023 by Xiaorui Sun.

References

* {{cite book , last=Johnson , first=D. L. , date=1997 , title=Presentations of Groups , edition=2nd , location=Cambridge , publisher=

Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

, doi=10.1017/CBO9781139168410 , isbn=0521372038 , page=49 Group theory Undecidable problems