The difference, of course, being that in semigroups we can "fall" inescapably from one J-class to another, but in groups every* element is in the same eggbox (compare transitivity with Boethius and the Wheel of Fortune).
Lagniappe: https://www.youtube.com/watch?v=-yvpRvd44s4&t=90s
* different identities are but a conjugation away?
(Shouldnt Pan the Smith be asleep here?)
(Galileo would have added Sagredo, somehow playing host to both Londoners and Arcadians?)
Interesting that these toga-clad truth-loving Arcadians predate Wonder Woman?
Lagniappe: https://images.squarespace-cdn.com/content/v1/51b3dc8ee4b051...
BTW, I messed up with the conjugation: of course conjugating an identity yields itself, but even if I don't recall it properly there is some way to get a torsor from a group by messing with "phase".
you might be more interested in torsion, though..
Ps: i think edwardian eras are a transitional period where, to stand apart from the previous era fetishizing wisdom (& the payment of dues), intellectuals began to focus their attention on the (often full of youth & signaling sprezzatura) acolytes of truth (& no taxes)
(and I'm interested in torsion, specifically how to hook together torsion groups via co-cycles, but haven't made much progress here over decades)
In principle there's a galois connection between wisdom and truth but in practice the adjoints are not easy to compute.