Quasinormal Subgroup¶

A subgroup that permutes, by set product, with every subgroup of its ambient group.

Version: v1 · 2026-10-31 · History
Domain-specific #: 14562
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Group Theory → Mathematics
Aliases: Permutable Subgroup

Core Idea¶

A subgroup H of a group G is quasinormal, also called permutable, when HK=KH for every subgroup K of G. HK and KH are sets of products, so this does not say every element of H commutes with every element of K.^[1]

The test is about the pair (H,G). Normal subgroups always pass it, but a subgroup can pass without being normal.^[1]

How would you explain it like I'm…

No faithful explanation at this level. All three generators agree that at a five-year-old level HK = KH collapses into 'order doesn't matter for each move' (elementwise commutation), whereas only the whole product sets must coincide, for every subgroup K.

Same-Either-Way Clubs

In math, a group is a set of moves, like turns and flips of a square, that you can combine. A subgroup H is a smaller club of moves that is itself a group. H is quasinormal if, for every other club K, doing 'an H move then a K move' gives exactly the same collection of results as doing 'a K move then an H move.' Single pairs of moves don't have to match in both orders; only the whole collections of results have to be the same. And it has to work for every club K, not just one.

Permutable Subgroups

For subgroups H and K of a group G, the set product HK is the set of all products hk with h in H and k in K. A subgroup H is quasinormal (or permutable) in G when HK = KH for every subgroup K of G. This is a statement about sets: hk need not equal kh, but every product formed one way can also be formed the other way with possibly different elements. Every normal subgroup is quasinormal, but some non-normal subgroups are quasinormal too, so this is a weaker condition than normality. It's enough to check K ranging over cyclic subgroups, and when the condition holds, HK is automatically the subgroup generated by H and K. Permuting with just one particular K is not enough.

A subgroup H ≤ G is quasinormal, or permutable, if HK = KH as sets for all subgroups K ≤ G. The equality is of product sets, not elementwise commutation, so hk may equal some other k'h'. Under this condition HK is itself a subgroup, namely the join ⟨H, K⟩. It suffices to verify the condition for cyclic subgroups K. Quasinormality is a property of the pair (H, G), not of H alone. It relaxes normality: every normal subgroup is quasinormal, and there are nonnormal subgroups that are still quasinormal. The universal quantifier is essential, since permuting with a single favored K is a much weaker statement.

Scope of Application¶

The name applies literally to subgroups inside a specified ambient group. It works for finite and infinite groups, but some consequences depend on finiteness: a finite-group quasinormal subgroup is subnormal, while an infinite-group example need not be.^[1]^[2]

In finite-group usage, an Iwasawa group is a different, group-level assertion: every subgroup in G is permutable.^[3] One quasinormal H does not make its G an Iwasawa group, as the S₃ example below shows.

Clarity¶

For subgroups H and K, HK means {hk : h∈H, k∈K}. Stonehewer shows that H is quasinormal if it passes the test even just against every cyclic K; when HK=KH, their product equals the generated subgroup ⟨H,K⟩.^[1]

Manages Complexity¶

The universal condition converts many separate subgroup-product checks into one stable property of H in G. A single K with HK≠KH rejects the claim; proving it for all cyclic K establishes it.^[1]

Abstract Reasoning¶

If H is quasinormal, any K can be joined to it by forming HK, which is already a subgroup. For a finite G this permits the additional inference that H is subnormal; do not carry that inference to arbitrary infinite G.^[1]^[2]

Knowledge Transfer¶

The same definition can classify H in very different groups. A nonnormal complement in a suitable 27-element cyclic-by-cyclic group and a normal subgroup A₃ in S₃ both pass for different reasons.^[1] Outside group theory the phrase is an analogy unless there are actual subgroups and set products.

Examples¶

Nonnormal example. Let G=C₉⋊C₃ with a⁹=b³=1 and bab⁻¹=a⁴, and H=⟨b⟩; this explicit presentation is derived from Stonehewer's odd-prime extension class.^[1] Every subgroup of this G permutes, while H is not normal because aba⁻¹∉H.^[1] The presentation and conjugation check are deductions from the source's class, not printed examples.^[1] Mapped back: H is the tested subgroup, G is the ambient group, and every K≤G gives HK=KH=⟨H,K⟩.^[1]

One-subgroup example. Let G=S₃ and H=A₃. H is normal and therefore quasinormal.^[1] But ⟨(12)⟩ and ⟨(13)⟩ have different products in opposite orders, so S₃ itself does not have every subgroup permutable; this is a direct calculation.^[1] Mapped back: H passes the all-K test while another pair of subgroups in the same G fails it.^[1]

Relationships to Other Abstractions¶

Current abstraction Quasinormal Subgroup Domain-specific

Parents (2) — outgoing typed relations

Quasinormal Subgroup is a kind of Closure Prime

Quasinormality is closure of Sub(G) under the unary operation K to HK for a fixed subgroup H.
Quasinormal Subgroup is a kind of Embedding Prime

A quasinormal subgroup is a faithful group embedding with universal subgroup-product permutability added.

Hierarchy paths (2) — routes to 2 parentless roots

Quasinormal Subgroup → is a kind of Closure

Show alternative path (1)

Neighborhood in Abstraction Space¶

Quasinormal Subgroup sits in a moderately populated region (59^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Algebraic Substructures & Closures (13 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-31

Not to Be Confused With¶

Normal subgroup: normality is sufficient, not necessary.
Iwasawa group: its every-subgroup claim is stronger than the one-H claim.
Elementwise commutation: HK=KH is equality of product sets; it can hold without hk=kh for each h,k.

References¶

[1] Stewart E. Stonehewer, “Old, Recent and New Results on Quasinormal Subgroups,” Irish Mathematical Society Bulletin 56 (2005), 125–133, especially p. 125. Full society-hosted PDF inspected: https://www.irishmathsoc.org/bull56/GiG5612.pdf The displayed C₉ ⋊ C₃ presentation and S₃ products are explicit deductions from the cited general rules and were independently recalculated; the source does not print those exact cases. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n

[2] John C. Lennox, “On quasinormal subgroups of certain finitely generated groups,” Proceedings of the Edinburgh Mathematical Society 26, no. 1 (1983), 25–28, especially the publisher's opening extract. Only the publisher extract was inspected, not the complete article: https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/on-quasinormal-subgroups-of-certain-finitely-generated-groups/BADAE01EE8F3C44F0C6AE97CBE5F7122 registry ↩a ↩b

[3] Marius Tărnăuceanu, “Minimal Non-Iwasawa Finite Groups,” Results in Mathematics 73, article 143 (2018), Introduction, p. 1. Full author-hosted journal PDF inspected: https://www.math.uaic.ro/~martar/pdf/articole/articol%2099.pdf The paper restricts its claims to finite groups and distinguishes Iwasawa groups from merely modular groups. registry ↩