Group isomorphism problem

In abstract algebra, the group isomorphism problem is the decision problem of determining whether two given finite group presentations present isomorphic groups.

The isomorphism problem was identified by Max Dehn in 1911^[1] as one of three fundamental decision problems in group theory; the other two being the word problem and the conjugacy problem. All three problems are undecidable: there does not exist a computer algorithm that correctly solves every instance of the isomorphism problem, or of the other two problems, regardless of how much time is allowed for the algorithm to run. In fact the problem of deciding whether a group is trivial is undecidable,^[2] a consequence of the Adian–Rabin theorem due to Sergei Adian and Michael O. Rabin.

References

^ Dehn 1911.
^ Miller, Charles (1992). "Decision problems for groups—survey and reflections." (PDF). Algorithms and classification in combinatorial group theory. Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989). Corollary 3.4: Springer. pp. 1–59.{{cite conference}}: CS1 maint: location (link)

Magnus, Wilhelm; Abraham Karrass; Donald Solitar (1976). Combinatorial group theory. Presentations of groups in terms of generators and relations. Dover Publications. p. 24. ISBN 0-486-63281-4.
Johnson, D.L. (1990). Presentations of groups. Cambridge University Press. p. 49. ISBN 0-521-37203-8.
Dehn, Max (1911). "Über unendliche diskontinuierliche Gruppen". Math. Ann. 71: 116–144. doi:10.1007/BF01456932.

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[FOOTNOTEDehn1911-1] Dehn 1911.

[2] Miller, Charles (1992). "Decision problems for groups—survey and reflections." (PDF). Algorithms and classification in combinatorial group theory. Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989). Corollary 3.4: Springer. pp. 1–59.{{cite conference}}: CS1 maint: location (link)

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