nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqqmshcslmwan3erj3ulczepe8q8a8xdyfkndnk8fudgtg8ccwg6zqlsl9jz (nprofile…l9jz) nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqt76as6gjr7pzg0taz40e55smjjegmj89ud7g056aqed90hs7cynsacyu7x (nprofile…yu7x) - I'm afraid this is one of those nasty puzzles where I don't know the answer. Worst case: the guy who confidently told me SmallGroup(32,43) has this surprising property was wrong, or it's not the group of affine transformations we think it is.
It's true that in a semidirect product A ⋉ B with both A and B abelian, conjugation by any element leaves the first component in (a,b) ∈ A ⋉ B unchanged. (Another fun example: the rotation/translation group of the plane.)
The only way out I see is what nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqqmshcslmwan3erj3ulczepe8q8a8xdyfkndnk8fudgtg8ccwg6zqlsl9jz (nprofile…l9jz) suggests: there's an automorphism f that multiplies b by 3,5,or 7 in a manner that depends nontrivially on a:
f(a,b) = (a, n(a)b)
where n(a) = 3,5, or 7.
Thanks for tackling this!
John Carlos Baez /
npub17u…8pd6m
2025-03-15 15:49:03
in reply to nevent1q…eyjp
Author Public Key
npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6mSeen on
wss://relay.mostr.pubPublished at
2025-03-15 15:49:03Event JSON
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"content": "nostr:nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqqmshcslmwan3erj3ulczepe8q8a8xdyfkndnk8fudgtg8ccwg6zqlsl9jz nostr:nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqt76as6gjr7pzg0taz40e55smjjegmj89ud7g056aqed90hs7cynsacyu7x - I'm afraid this is one of those nasty puzzles where I don't know the answer. Worst case: the guy who confidently told me SmallGroup(32,43) has this surprising property was wrong, or it's not the group of affine transformations we think it is.\n\nIt's true that in a semidirect product A ⋉ B with both A and B abelian, conjugation by any element leaves the first component in (a,b) ∈ A ⋉ B unchanged. (Another fun example: the rotation/translation group of the plane.)\n\nThe only way out I see is what nostr:nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqqmshcslmwan3erj3ulczepe8q8a8xdyfkndnk8fudgtg8ccwg6zqlsl9jz suggests: there's an automorphism f that multiplies b by 3,5,or 7 in a manner that depends nontrivially on a:\n\nf(a,b) = (a, n(a)b)\n\nwhere n(a) = 3,5, or 7.\n\nThanks for tackling this!",
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