Imperfect Group¶
A group with no nontrivial perfect quotient, including itself among the quotients tested.
Core Idea¶
An imperfect group is a group with no nontrivial perfect quotient. A quotient is perfect when it equals its own commutator subgroup. Berrick and Robinson use this stronger condition rather than merely saying that the starting group is not perfect.[^ref-f89c84ecbcf3]
For every normal subgroup N of G, test the quotient G/N. The quotient G/1=G counts if G is nontrivial. One nontrivial perfect quotient excludes G; the trivial quotient G/G does not. The trivial group itself qualifies vacuously.[^ref-f89c84ecbcf3]
Scope of Application¶
This is a group-theoretic class defined through all normal quotients, with no finite-order assumption in the definition. The worked cases S3 and S5 are finite because their complete quotient lists can be checked. The source paper's publisher abstract supports the definition, while university notes support the elementary example calculations. Broader closure and embedding claims in the seed are not asserted.[ref-f89c84ecbcf3][ref-4cb6c9738ab5]
Clarity¶
Not perfect checks G alone. Imperfect checks every nontrivial quotient of G, including G itself. A5 × C2 is not perfect because it maps onto abelian C2, but it is not imperfect: first projection maps it onto nontrivial perfect A5.[^ref-4cb6c9738ab5]
A perfect subgroup is different from a perfect quotient. S5 contains perfect normal A5, but A5 is not a quotient of S5. Its quotient list is S5, C2 and 1; neither nontrivial member is perfect.[^ref-4cb6c9738ab5]
Manages Complexity¶
A complete normal-subgroup list turns the universal condition into a manageable quotient inventory. For S5, the source proves the list 1, A5, S5. The quotient list then contains only S5, C2 and 1. A negative decision needs less work: one surjection onto a nontrivial perfect group suffices.[^ref-4cb6c9738ab5]
Abstract Reasoning¶
For each normal N, ask whether nontrivial G/N equals [G/N,G/N]. G is imperfect exactly when the answer is no every time. Equivalently, each nontrivial quotient must have some nontrivial abelian quotient of its own. Checking only whether G has one abelian quotient would miss the A5 × C2 near miss.[ref-f89c84ecbcf3][ref-4cb6c9738ab5]
Knowledge Transfer¶
The same quotient test accepts solvable S3 and non-solvable S5. Solvability is therefore not the definition. What transfers between examples is the explicit group carrier, complete quotient family, perfectness test and universal exclusion. Outside group theory, a similar universal image rule is only an analogy unless those literal roles exist.[^ref-4cb6c9738ab5]
Example¶
S3, solvable positive case. Its normal subgroups 1, A3 and S3 give S3, C2 and 1. S3 has the sign quotient C2, and C2 is abelian; neither nontrivial quotient is perfect. Its derived series S3 ⊃ A3 ⊃ 1 ends, so this positive example is solvable.[^ref-4cb6c9738ab5]
S5, non-solvable positive case. The proved normal-subgroup list 1, A5 and S5 again gives S5, C2 and 1. Sign shows S5 is nonperfect and C2 is abelian, so all nontrivial quotients are nonperfect. Yet A5 is nonabelian simple and hence perfect inside S5; its presence makes S5 non-solvable without giving a perfect quotient.[^ref-4cb6c9738ab5]
Boundary: A5 × C2 fails. Its C2 image shows the product is nonperfect, but its A5 image is perfect.[^ref-4cb6c9738ab5]
Relationships to Other Abstractions¶
Current abstraction Imperfect Group Domain-specific
Parents (1) — more general patterns this builds on
-
Imperfect Group is a kind of Group Prime
Every imperfect group satisfies the Group axioms and additionally excludes every nontrivial perfect quotient.
Hierarchy paths (5) — routes to 5 parentless roots
- Imperfect Group → Group → Monoid → Semigroup → Set and Membership
- Imperfect Group → Group → Monoid → Identity Element
- Imperfect Group → Group → Monoid → Semigroup → Closure
- Imperfect Group → Group → Monoid → Semigroup → Associativity → Invariance
- Imperfect Group → Group → Monoid → Semigroup → Associativity → Symmetry
Neighborhood in Abstraction Space¶
Imperfect Group sits in a sparse region of the domain-specific corpus (68th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Group-Theoretic Structures & Subgroups (41 abstractions)
Nearest neighbors
- Strictly simple group — 0.85
- Perfect core — 0.84
- Thompson factorization — 0.84
- Normal closure (group theory) — 0.83
- Cyclic number (group theory) — 0.83
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
Nonperfect group: only G itself fails perfectness; it may have another perfect image. Solvable group: a narrower familiar positive pattern; S5 is imperfect and non-solvable. Perfect subgroup or Perfect Core: an internal group structure, not a quotient. The live Group Prime is the strict broader genus in the reviewed DAG; this entry adds the no-perfect-quotient condition.[^ref-4cb6c9738ab5]
References¶
[^ref-f89c84ecbcf3]: A. J. Berrick and D. J. S. Robinson, “Imperfect groups”, Journal of Pure and Applied Algebra 88 (1993): 3–22. The publisher abstract explicitly states the definition; the full paper was not verified for this entry.
[^ref-4cb6c9738ab5]: University of Washington, “Chapter 1 Group theory”, course notes hosted on S. Paul Smith’s teaching page, undated; Lemma 3.7 and Propositions 3.8 and 3.10 (PDF pp. 8–9). The sign map, S5 normal-subgroup list and A5 simplicity are sourced there; S3 and direct-product consequences are elementary deductions.