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a few finite group thingies (mainly reportative, very little personal research)
march 24th, 2023i'm more of an analyst, but i like giving algebra a go from time to time. this is one such instance. so, i was mainly thinking about finite group invariants lately, and notable elements of groups sideways through. for those who don't know, a finite group invariant is simply a function that yields the same output between two isomorphic finite groups (i'll just say groups from there on, by laziness). more feasibly, you might want to construct functions over something we can formalize as
however, invariants can be very dumb ; take ψ : S/≃ ⟶ {0} for instance, then you can build ψ* : S ⟶ im(ψ) = {0} such that... well, you don't really need to add much here, thankfully. however, since that just lumps every group in the same basket, it's not a very "useful" invariant. so, what do we mean by "useful" here ? well, we basically mean that, we want to be able to figure the isomorphism class of the group out of the invariant's output. so, basically, we want an invertible invariant (well, rather, invertible ψ, rather than ψ* ; invertible up to isomorphism, at best), which is what we more commonly call a characteristic invariant. obviously though, this is quite hard and can often yield fairly unwieldy invariants for larger groups, so a weaker yet interesting variant (pun intended) comes from "locally invertible" invariants (up to isomorphism) ; by that, i mean that, for ψ-1(im ψ) ⊇ A/≃, and ψ* : G ⟼ ψ([G]), there exists Γ ⊆ S/≃ such that ψ|Γ is bijective. we can notate such a Γ which we'll notate Γψ for an invariant ψ*. the idea is to explicate a Γψ as large as possible for a given invariant ψ*. you can also take n invariants ψ1*, ..., ψn* and create a new invariant ψ* : G ⟼ (ψ1*, ..., ψn*)(G) with a possible Γψ as ⋃i Γψi, which explicitly gives us a way to construct an increasing sequence of Γs, aka invariants that extend their usefulness to increasingly larger chunks of groups (up to isomorphisms) !... though not strictly, obviously.
since we mainly need to preemptively construct ψ a function over S/≃, since S/≃ is countably infinite, we can pull any invariant through a sort of indexer that maps their image injectively over ℕ, so from now on we'll just use ψ ∈ (S/≃)ℕ (yeah, sorry for staying in such ill-behaved spaces for so long, my analyst side is showing up i guess lol).
also, since we need to preemptively construct ψ ∈ (S/≃)ℕ because we can get its corresponding invariant ψ*, this means we need to take things that we know for a fact exist in [G] in general and not just one specific G in its isomorphism class, whether these are elements or specific behaviors. the first exemple i thought of was ψ* : (G, ·) ⟼ #{g · g : g ∈ G}, since this quantity doesn't change by simple relabeling of elements and/or of the operator (which is exactly what an isomorphism is, anyway). this basically counts how many different elements there are in the diagonal in the group's cayley table (i think it's clearer why it works, then). another one, with a similar reasoning, is ψ* : (G, ·) ⟼ #{g · g = 0(G, ·) : g ∈ G}. none of these are characteristic, at least the former works for all groups up to order 4 (i'm too lazy to check back if it works for larger groups, but i remember it's around there). so i thought i'd try to construct more notable elements.
so, if you know about ring theory, there's this idea of ring characteristic, which is basically just a number, more precisely a natural number (i include 0, cope and seethe), κ ∈ ℕ such that κℕ = {n ∈ ℕ : 0 = 1+...+1, n times}. so i thought, i'd try to figure a way to define "1" this way, or rather succ(eG). so, expectedly i didn't come to an unique element, rather i say that succ(eG) is an element of G chosen out of its elements of greatest order. i didn't do much yet with that, but i thought i was neat lol.
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