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The relation "has the same order of" is of equivalence in the set of all the groups. The relation "is isomorphic to" is of equivalence in each set made of the groups of one same order (so, in a sense, it gets a finer partition of the set of all the groups). The equivalence relation "is affinely isomorphic to" is equivalent to "being isomorphic to", so it doesn't add anything from this standpoint. So, my question is: are there "finer/coarser" equivalence relations than equipotency/isomorphicity ones, which we may define on the set of all the groups?

Edit. Agreed from the comments, equality is the finest.

Kan't
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    I doubt there exists a set of all groups. If you take all subgroups of a given group, conjugacy is finer than isomorphism, and equality is the finest. – Christophe Boilley Feb 19 '25 at 12:44
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    Equality is the finest equivalence relation, and it is always defined. But it uses "non-group-theoretic data", i.e. it distinguishes groups based on properties of their elements that have nothing to do with the group operations. If you don't allow this, i.e. you require isomorphic groups to be considered equivalent, then group isomorphism itself is the finest equivalence relation. – Karl Feb 19 '25 at 13:29
  • In other words, isomorphism is the finest equivalence relation in univalent foundations, because it coincides with equality. – Naïm Camille Favier Feb 19 '25 at 16:08

2 Answers2

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One can define lots of equivalence relations on groups, based on various properties of the groups. The question is more what you want to do with them and which useful information you get from the equivalence relation. Some examples:

Let $[G,G]$ denote the commutator of the group $G$, then we can define $G \sim H$ when $[G,G]$ is isomorphic to $[H,H]$, one could combine that with the requirement that G and H have the same order. One could also just ask that the commutators have the same cardinality.

One could look at the highest order of an element in $G$ and define two groups to be equivalent if their highest order elements have the same order or if the subgroup generated by the elements of highest order is isomorphic.

One could look at the automorphism group of the group and define equivalence relations based on isomorphic automorphism groups or just automorphism groups of the same cardinality.

The possibilities are endless but you need some purpose why you want to sort all groups into some kind of boxes, some meaning attached to your boxes.

quarague
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There is another way to approach this: in 1940 Philip Hall (The classification of prime-power groups, appeared in the Journal für die reine und angewandte Mathematik, 182, pp. $130-141$) introduced the concept of isoclinism, an equivalence relation on groups coarser than isomorphism (being isomorphic implies being isoclinic, but not vice versa). The concept of isoclinism was introduced to classify $p$-groups, although the concept is applicable to all groups. Isoclinism can be extended to isologism, which is similar to isoclinism, but then w.r.t. a variety of groups. There exists a vast literature on isoclinism and isologism.

The isoclinism class of a group $G$ is determined by the groups $G/Z(G)$ ($G$ mod its center) and $G′$ (the commutator subgroup) and the commutator map from $G/Z(G) \times G/Z(G) \rightarrow G′$ (where $(\overline{x},\overline{y}) \mapsto [x,y]$). Two groups $G_1$ and $G_2$ are isoclinic if there are isomorphisms from $G_1/Z(G_1) \rightarrow G_2/Z(G_2)$ and from $G_1′ \rightarrow G_2′$ commuting with the commutator map. As an example, all abelian groups are isoclinic to each other, in fact they are isoclinic to the trivial group.

Nicky Hekster
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