Subgroup Isomorphism Problem

• The Subgroup Isomorphism Problem is defined by asking if an isomorphic copy of a finite group in the unit group of an integral group ring necessarily embeds into the original group.
• Structural analyses employ prime and N-prime graph invariants to detect subgroup embedding patterns and expose asymmetries in normalizer actions.
• Methodological advances like the HeLP method and combinatorial techniques have confirmed SIP in key group families while highlighting remaining open challenges.

The Subgroup Isomorphism Problem (SIP) is a central question in group theory and related algebraic structures, concerning whether and to what extent the existence of an abstractly isomorphic copy of a finite group $T$ embedded in some algebraic construction over a group $G$ guarantees that $T$ itself embeds as a subgroup of $G$. The most prominent formulation arises for the group $V(\mathbb{Z}G)$ of normalized units in the integral group ring $\mathbb{Z}G$ of a finite group $G$. The question is closely connected to the arithmetic, combinatorial, and representation-theoretic structure of $G$, and has deep links to the study of prime graph invariants, unit group theory, and algorithmic group theory.

1. Formal Statement and Variants

The classical SIP for integral group rings asks: given a finite group $G$ and a finite group $T$, if there exists an embedding $T \hookrightarrow V(\mathbb{Z}G)$, is it necessarily true that $T \hookrightarrow G$? In other words, does the occurrence of an isomorphic copy of $T$ in the unit group structure always reflect an actual subgroup inside $G$ itself? This formulation is motivated by the property that $V(\mathbb{Z}G)$ contains $G$ via group elements, but may also contain more complicated or "exotic" finite subgroups not present in $G$.

Formally,

$\text{If } T \leq V(\mathbb{Z}G) \text{ (i.e., %%%%17%%%% contains a subgroup isomorphic to %%%%18%%%%)}, \text{ must } T \leq G \text{ (i.e., isomorphic to an actual subgroup of %%%%19%%%%)?}$

This problem admits natural analogs and generalizations in various settings:

• Isomorphic subgroups within topological or algebraic group completions.
• Transfer of subgroup isomorphism through other functorial group constructions or posets of subgroups (cf. poset-theoretic approaches (Tarnauceanu, 2015)).
• Algorithmic versions in finitely presented groups or classes such as mapping class groups or right-angled Artin groups (Koberda, 2010).

2. Structural and Graph-Theoretic Invariants

The SIP is deeply intertwined with group invariants that encode the subgroup structure, notably the classical Gruenberg–Kegel (prime) graph ${\rm GK}(G)$ and its refined directed analog, the N-prime graph $\Gamma_{N(G)}$ (Pacifici et al., 3 Nov 2025).

• Gruenberg–Kegel graph: Vertices are the set of primes $\pi(G)$ dividing orders of elements of $G$; undirected edges $p$–$q$ appear if $G$ contains an element of order $pq$.
• N-prime graph $\Gamma_{N(G)}$: Retains $\pi(G)$ as vertices; a directed arc $q \rightarrow p$ exists iff $N_G(C_p)$ (normalizer of a subgroup of order $p$) contains elements of order $q$. This directed structure captures asymmetries in subgroup normalizers and refines ${\rm GK}(G)$, providing a sensitive invariant for subgroup embedding phenomena:

$$q \rightarrow p \iff \exists\, a \in G, |a| = p, \exists\, y \in N_G(\langle a \rangle), |y| = q.$$

Unit group compatibility: By a theorem of Cohn–Livingstone, $\pi(V(\mathbb{Z}G)) = \pi(G)$. It's always the case that $\Gamma_{N(G)} \subseteq \Gamma_{N(V(\mathbb{Z}G))}$.

The main structural question—whether $\Gamma_{N(V(\mathbb{Z}G))} = \Gamma_{N(G)}$ (the NPQ or N-prime Graph Question)—forms a close parallel to the SIP and can be used as a powerful tool for its study (Pacifici et al., 3 Nov 2025).

3. Advances and Results in the Integral Group Ring Case

Major progress on the SIP for $V(\mathbb{Z}G)$ includes the following established results:

• Cyclic and elementary abelian cases: SIP holds for cyclic groups of prime power order and for $C_p \times C_p$ (Margolis, 2015). The cyclic case follows from the exponent result of Cohn–Livingstone; the elementary abelian case from partial augmentation and character-theoretic constraints (HeLP method).
• New families: SIP is affirmed for $C_4 \times C_2$ and for all $2$-subgroups when $G$ has a dihedral Sylow-$2$ subgroup (Margolis, 2015). Proof involves reduction via group-theoretic classification (Gorenstein–Walter, Alperin–Brauer–Gorenstein), combinatorial subgroup analysis, and the character-theoretic HeLP method.

A table outlining key cases and results is given below:

$T$	$G$ Condition	SIP Holds?	Source
$C_n$ (cyclic p-power)	Arbitrary	Yes	(Margolis, 2015)
$C_p \times C_p$	Odd $p$	Yes	(Margolis, 2015)
$C_4 \times C_2$	Arbitrary	Yes	(Margolis, 2015)
$2$-subgroup	Dihedral Sylow-$2$ in $G$	Yes	(Margolis, 2015)
Frobenius $C_p \rtimes C_{q^k}$	$G$ solvable	Yes	(Pacifici et al., 3 Nov 2025)

Notably, by (Pacifici et al., 3 Nov 2025), for finite solvable $G$, whenever $V(\mathbb{Z}G)$ contains a Frobenius subgroup $C_p \rtimes C_{q^k}$, $G$ itself embeds such a subgroup.

Counterexamples to general SIP are also known: Hertweck constructed nonisomorphic $G$, $H$ with $\mathbb{Z} G \cong \mathbb{Z} H$, so that $H$ embeds in $V(\mathbb{Z}G)$ but not in $G$ directly.

4. Methodological Innovations

Two major methodological directions inform the contemporary study of SIP:

1. Prime graph and N-prime graph invariants: The refined analysis via $\Gamma_{N(G)}$ enables reduction of SIP and NPQ to smaller building blocks, especially almost simple groups. Directed edge structure encodes more subgroup-theoretic data, distinguishing cases where undirected graphs coincide but group-theoretic embedding properties differ (Pacifici et al., 3 Nov 2025).
2. HeLP method (Herstein–Livingstone–Passi): Uses partial augmentations and character theory to relate group ring units to group elements. Key ingredients involve computing constraints on possible partial augmentations (via class functions and character values) for torsion units, establishing rational conjugacy in $G$, and ruling out existence of units with group-theoretically forbidden properties (Margolis, 2015).

Additionally, combinatorial arguments exploiting bipartite graphs of normalizer actions in 2-Frobenius situations and careful analysis of metacyclic extension structures are employed to realize Frobenius subgroups or their generalizations (Pacifici et al., 3 Nov 2025).

5. Algorithmic and Poset-Theoretic Aspects

The subgroup isomorphism question generalizes to algorithmic and poset settings:

• Algorithmic SIP in mapping class groups and RAAGs: The problem is undecidable in mapping class groups of sufficiently complex surfaces, due to the embedding of unsolvable groups like $F_2 \times F_2$ (Koberda, 2010). For right-angled Artin groups (RAAGs), SIP becomes decidable by reconstructing the defining graph from cohomological invariants.
• Poset of isomorphism classes $\Iso(G)$: Studies the structure of the poset of isomorphism classes of subgroups, ordered by inclusion up to isomorphism. This poset captures information about possible subgroup isomorphism types and, in some settings (e.g., abelian $p$-groups, ZM-groups), fully determines the group up to isomorphism (Tarnauceanu, 2015). The congruence between $\Iso(G)$ and the lattice of subgroups may fail in general, reflecting the nontriviality of SIP.

6. Extensions, Limitations, and Open Problems

A number of challenges and open directions remain:

• Extension to nonsolvable groups: SIP for $V(\mathbb{Z}G)$ and the NPQ are open for arbitrary groups outside the families of alternating, $\operatorname{PSL}_2(p^f)$ ($f \le 2$), and rational groups. Counterexamples in the general isomorphism problem provide obstructions in the largest possible cases (Pacifici et al., 3 Nov 2025).
• Metacyclic and higher-rank cases: Partial results exist when $G'$ is cyclic or for certain metacyclic subgroups, but no general solution.
• Lifting embeddings through quotients: It is not generally possible to lift a subgroup embedding $C_p \rtimes C_{q^2} \hookrightarrow G/M$ to an embedding in $G$, revealing obstructions related to the subgroup lattice and cohomological structure.
• Algorithmic classification and descriptive set-theoretic complexity: For oligomorphic groups, the equivalence of isomorphism is Borel reducible to a countable Borel equivalence relation; essentially, the isomorphism problem is "tame" in the descriptive set-theoretic hierarchy (Nies et al., 2019).

Open problems include systematizing which $T$ (such as more general Frobenius or metacyclic groups) satisfy SIP for all finite $G$, clarifying the role of the N-prime graph for broader classes, and resolving the classification of group constructions by their posets of isomorphism classes.

7. Examples and Illustrative Cases

Selected examples from (Pacifici et al., 3 Nov 2025) and (Margolis, 2015) illustrate the subtleties of SIP:

• Directed prime graph distinction: $\operatorname{PSL}_2(7)$ and $\operatorname{PSL}_2(8)$ share the same (undirected) prime graph but differ as directed N-prime graphs, affirming the directed variant's greater sensitivity to normalizer structure.
• Frobenius extension in solvable group: If $V(\mathbb{Z}G)$ embeds $C_p \rtimes C_{q^k}$ for $G$ solvable, $G$ contains such a subgroup. However, in quotient situations, embeddings may not always lift.
• 2-Frobenius group: $G=(C_3^6\rtimes C_7)\rtimes C_3$ and its Frobenius subgroup have matching disconnected prime graphs, but the N-prime graph detects a unique directed edge related to the semidirect product structure.

These cases demonstrate how SIP and related invariants provide both structural insight and practical criteria for subgroup realizability in algebraic contexts.