12.2 The isomorphism problem

The isomorphism problem for groups is a fundamental challenge in group theory, asking if an algorithm exists to determine whether two groups are isomorphic. It's crucial for understanding group structure and has applications in cryptography and coding theory. The problem is known to be in NP but its exact complexity remains open.

Solving the isomorphism problem involves finding a bijective function between groups that preserves operations. While some special cases are solvable in polynomial time, the general problem for finite groups remains unsolved. This highlights the deep connections between group theory, computational complexity, and algorithmic challenges in mathematics.

Isomorphism problem for groups

Definition and significance

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• Isomorphism problem for groups asks whether an algorithm exists to determine if two given groups are isomorphic
• Group isomorphism consists of a bijective homomorphism between two groups preserving group structure and operations
• Fundamental in group theory with significant implications for understanding group structure and classification
• Closely related to the study of group invariants and development of computational group theory
• Importance extends to various areas of mathematics and computer science (cryptography, coding theory)
• Known to be in NP (nondeterministic polynomial time) but not known to be NP-complete or in P (polynomial time)

Mathematical formulation and properties

• Formally defined as finding a bijective function $f: G \rightarrow H$ between groups G and H such that $f(ab) = f(a)f(b)$ for all $a, b \in G$
• Preserves group operations including identity element, inverses, and subgroup structure
• Isomorphic groups have the same order, element orders, and number of elements of each order
• Isomorphism induces a one-to-one correspondence between subgroups of G and H
• Cayley's theorem demonstrates every group is isomorphic to a subgroup of a symmetric group
• Examples of isomorphic groups include $(\mathbb{Z}, +)$ and $(2\mathbb{Z}, +)$ (integers under addition and even integers under addition)

Applications and related concepts

• Used in classification of finite simple groups, a major achievement in 20th-century mathematics
• Critical in understanding symmetry groups in physics and chemistry (point groups, space groups)
• Applied in cryptography for analyzing security of group-based cryptosystems (elliptic curve groups)
• Connects to representation theory through character tables, which are invariant under isomorphism
• Utilized in computer algebra systems for efficient group computations and simplifications
• Related to the study of automorphism groups, which describe self-isomorphisms of a group
• Examples of applications include determining equivalence of chemical compounds (graph isomorphism) and analyzing social networks (group actions)

Challenges in group isomorphism

Computational complexity

• Difficulty increases exponentially with group size, making brute-force approaches impractical for large groups
• No known polynomial-time algorithm for general finite groups
• Best known algorithm for finite groups has complexity $n^{\log n + O(1)}$ where n is the group order
• Special cases like abelian groups have polynomial-time algorithms (structure theorem for finitely generated abelian groups)
• Graph isomorphism reducible to group isomorphism, connecting its complexity to a well-studied problem
• Memory requirements for storing large group structures can be prohibitive (multiplication tables grow quadratically)

Structural challenges

• Different group presentations can represent isomorphic groups (multiplication tables, generating sets, relator sets)
• Presence of automorphisms within groups creates multiple valid isomorphisms, complicating the search for a single solution
• Determining isomorphism often requires deep understanding of group-theoretic properties and structure
• Challenge of efficiently computing and comparing group invariants (element orders, subgroup structure)
• Infinite groups pose additional challenges, with the problem being undecidable in general
• Examples of challenging cases include distinguishing between non-isomorphic groups of the same order (Q8 and D8, both of order 8 but non-isomorphic)

Algorithmic approaches and limitations

• Nauty algorithm for graph isomorphism adapted for certain classes of groups, but not generally applicable
• Canonical form algorithms attempt to create unique representations, but struggle with large symmetry groups
• Coherent configuration methods show promise for certain group classes, but have limitations
• Quantum algorithms proposed, but practical implementation remains challenging
• Heuristic approaches like randomly generating and testing isomorphisms can be effective for small groups
• Machine learning techniques explored for isomorphism testing, but struggle with generalization to arbitrary groups
• Examples of successful algorithms include the p-group generation algorithm for groups of prime power order

Solvability of the isomorphism problem

Polynomial-time solvable cases

• Abelian groups solvable in polynomial time using structure theorem for finitely generated abelian groups
• Finite p-groups (groups of prime power order) solvable using nilpotent quotient algorithm
• Finitely generated nilpotent groups solvable through combination of abelian and p-group algorithms
• Certain classes of solvable groups with bounded derived length have polynomial-time solutions
• Groups with bounded genus (can be drawn on a surface of fixed genus without edge crossings) are polynomial-time solvable
• Examples include cyclic groups, elementary abelian groups, and dihedral groups of order 2n

Decidable but potentially non-polynomial cases

• Polycyclic groups have decidable isomorphism problem, but not known to be in polynomial time
• Hyperbolic groups have decidable isomorphism problem using Dehn's algorithm and normal forms
• One-relator groups with torsion have decidable isomorphism problem (Magnus' solution)
• Automatic groups have decidable isomorphism problem using finite state automata techniques
• Certain classes of metabelian groups have decidable isomorphism problems
• Examples include Baumslag-Solitar groups and certain classes of HNN extensions

Undecidable and open cases

• Isomorphism problem undecidable for general infinite groups (existence of finitely presented groups with undecidable word problem)
• Problem remains open for general finite groups, with no known polynomial-time algorithm
• Undecidable for arbitrary finitely presented groups (reduction from the word problem)
• Open for certain classes of solvable groups and groups of intermediate growth
• Complexity status unknown for many important classes of groups (simple groups, matrix groups)
• Examples of groups with undecidable isomorphism problem include certain constructions based on Adian-Rabin groups

Isomorphism problem vs other group theory problems

Relationship to recognition and classification problems

• Closely related to group recognition problem (determining if a group is isomorphic to a specific known group)
• Connected to the group classification problem (determining structural properties of a group)
• More difficult than determining basic group properties (order, abelian/non-abelian)
• Related to the automorphism group problem (finding all self-isomorphisms of a group)
• Connects to the problem of finding canonical forms for groups
• Examples include distinguishing between simple groups of the same order (A5 and PSL(2,5))

Comparison with other decidability problems

• Unlike word problem, isomorphism problem remains open for general finite groups
• Conjugacy problem (determining if two elements are conjugate) generally easier but still challenging for many group classes
• Subgroup membership problem decidable for finite groups but undecidable for some infinite groups, similar to isomorphism problem
• Isomorphism problem for graphs reducible to group isomorphism, establishing connection to graph theory
• Generally more difficult than computing group-theoretic invariants (center, derived series, composition series)
• Examples of related problems include the centralizer problem and the normalizer problem in groups

Computational complexity comparisons

• Computational complexity generally higher than other common group-theoretic problems
• NP-hard for certain classes of groups, but not known to be NP-complete for general finite groups
• Space complexity can be significant, often requiring storage of full group structure
• Requires comparison of two distinct algebraic structures, unlike problems focused on a single group
• Harder than problems solvable by standard group algorithms (Todd-Coxeter, Knuth-Bendix)
• Examples of complexity differences: computing the order of a permutation group is in P, while group isomorphism is not known to be in P