Subgroups of S∞

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H14All independent pair flipsClosed, not open

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Cycle length 2

g(n) 3g⁻¹(n) 3

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0–79One member of H

Mathematics

H14 · SUBGROUP OF S∞

All independent pair flips

The full subgroup, its members, and its topology.

THE CURRENT DRAWING

Flip pair j when j is not divisible by 2.

The canvas draws $g=r^{1}$ , where r is this representative.

This is one member, not an enumeration of the subgroup.

DEFINITION & EXPLANATION

Definition

$\{g:g(B_j)=B_j\text{ for every }j\}$

Here $B_j=\{2j,2j+1\}$ is the jth pair.

Structure and topology

Independent pair flips, isomorphic to $\prod_j C_2$ . A compact closed group. Every chosen set of flipped pairs, including infinite choices, defines a member.

Closure in $S_\infty$ : The subgroup itself.

TRY THIS

Inspect the relation each member preserves. Try another member and compare. A finite violation excludes an outsider; no finite picture alone establishes every defining condition.

Proofs, notation & background+

Finite observations and the topology

A basic neighborhood is

U(g,F)=\{h\in S_\infty:h|_F=g|_F\},\qquad F\subseteq\mathbb N\text{ finite}.

Thus $g_j\to g$ in $S_\infty$ exactly when, for each $n$ , the values $g_j(n)$ eventually equal $g(n)$ . The limit in this assertion is itself a permutation. A pointwise limit in the larger function space $\mathbb N^{\mathbb N}$ can fail to be surjective and hence lie outside $S_\infty$ .

Write $\operatorname{Fix}(F)=\{g:g(x)=x\text{ for every }x\in F\}$ and $\operatorname{Stab}(F)=\{g:g(F)=F\}$ . The groups $\operatorname{Fix}(F)$ for finite $F$ form an identity-neighborhood basis. This is the setup used in KPT §4, and in Cameron, §2.5.

For a subgroup $H\le S_\infty$ :

H\text{ open}\iff\exists F\text{ finite}:\operatorname{Fix}(F)\leq H.

Indeed, an open $H$ contains a basic identity neighborhood; conversely, if it contains $\operatorname{Fix}(F)$ , it is a union of open cosets of that subgroup.

Also,

H\text{ closed}\iff \forall g\notin H\ \exists F\text{ finite}: U(g,F)\cap H=\varnothing.

Equivalently, $g$ belongs to the closure of $H$ exactly when every finite restriction of $g$ extends to some member of $H$ . Therefore openness gives finite certificates for membership at members; closedness gives finite certificates for nonmembership at outsiders. These are statements about full permutations, not finite subsets of $\mathbb N$ being themselves open or closed.

An open subgroup is always closed. Every other coset is open, and their union is its complement. The possible categories are consequently: both open and closed; closed but not open; and neither. There is no open-but-not-closed subgroup of $S_\infty$ .

Proof tools for the closed table

Finite constraints and relations. Failure to preserve a finitary relation is witnessed by one finite tuple. The neighborhood fixing that tuple’s images contains no automorphism. Coordinate equalities defining H11–H18 and H26 likewise have finite witnesses to failure. For example, failure to preserve a named row is seen on one arrow; failure to preserve an unnamed partition can be witnessed by a pair whose equivalence relation changes.

Not open by an unseen violation. For H11–H18 and H26, after fixing any finite $F$ , choose fresh points or fresh blocks and make a transposition violating the defining condition. For an unnamed pair partition, exchange one point of each of two fresh pairs; their partners remain fixed, so pair equivalence fails. For an unnamed infinite-row partition, exchange one point between two fresh rows while leaving other points of both rows fixed. These permutations belong to $\operatorname{Fix}(F)$ , so no $\operatorname{Fix}(F)$ lies within the subgroup.

For H23–H24, exchanging two unobserved rational points reverses their order. For H25, the graph’s extension property supplies fresh vertices $a,b$ with different adjacency to a third vertex $c$ , so exchanging only $a$ and $b$ violates an edge or nonedge while fixing the prescribed finite $F$ . The graph’s homogeneity also shows nondiscreteness: outside any finite $F$ there are distinct vertices with the same adjacency pattern to $F$ , and a finite isomorphism moving one to the other extends to an automorphism.

For H24, both denominator classes really are dense. Fractions with odd denominator are dense by using arbitrarily large odd denominators; reducing them preserves oddness of the denominator. Dyadic fractions with odd numerator and denominator $2^k$ , $k\ge1$ , witness density of the even-denominator class. Back-and-forth now extends finite order-and-label-preserving maps, giving nontrivial automorphisms fixing any finite set. H23 is its one-class analogue. The rational groups have infinite point orbits, as does H25.

Finite bases. If a finite $F$ has trivial pointwise stabilizer in $H$ , then $H$ is discrete and closed in $S_\infty$ . To prove closedness, let $u$ lie in the closure of $H$ . Some $h\in H$ agrees with $u$ on $F$ . This $h$ is unique, because two such elements would have quotient fixing $F$ . For any further $n$ , closure supplies an element agreeing with $u$ on $F\cup\{n\}$ ; uniqueness forces that element to be $h$ . Thus $u(n)=h(n)$ for every $n$ and $u=h\in H$ . This proves the assertions for H19–H22. Their countability proves nonopenness: every $\operatorname{Fix}(F)$ contains uncountably many independent choices of swaps on fresh pairs.

Compactness. A closed subgroup of $S_\infty$ is compact exactly when all its point orbits are finite. In one direction, each evaluation map sends a compact group into the discrete set $\mathbb N$ and must have finite image. In the other direction, a closed group with finite point orbits is a closed subgroup of the product of the finite symmetric groups on those orbits, hence compact. This proves compactness of H14 and H26, as well as noncompactness of the listed groups with infinite point orbits. Finite groups are compact too.

Thus “closed” includes finite groups, infinite compact groups, infinite discrete groups, and nondiscrete noncompact groups. They should not share a single geometric stereotype.

Closed groups as automorphism groups

There is a general representation theorem: every closed subgroup of $S_\infty$ is the automorphism group of a countable relational structure on $\mathbb N$ . One can use a relation for each orbit on finite tuples. Conversely, automorphism groups of finitary relational structures are closed by the finite-violation argument. KPT §2 gives this correspondence.

This describes all closed subgroups conceptually; it does not turn them into a short list of formulas. For broader research context, Bergman and Shelah, Closed subgroups of the infinite symmetric group, studies a classification under a specified equivalence relation on subgroups. That is not a claim that only four closed subgroups or four isomorphism types exist.

The ambient group matters

If $G$ is one of the closed nonopen examples and $F$ is finite, then

G_{(F)}=G\cap\operatorname{Fix}(F)

is open and closed in $G$ , but is not open in $S_\infty$ . If it were open in $S_\infty$ , the overgroup $G$ would also be open as a union of its cosets. It is closed in $S_\infty$ because $G$ and $\operatorname{Fix}(F)$ are closed.

For example, pinning finitely many points within the parity-preserving group H12 gives an open subgroup of H12, while infinitely many parity conditions still prevent ambient openness. A subgroup display must therefore always identify its ambient group.

FURTHER READINGCameron · Infinite permutation groups Kechris–Pestov–Todorcevic

AI-assisted mathematical commentary. Definitions and proof arguments describe infinite objects; finite computations illustrate them.