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Loops and their multiplication groups
A thread in 15 parts
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A loop is a quasigroup with an identity element. The story of why they are called loops is an interesting one and may even be true, but I will save it for another day. I am going to focus on loops in this thread.
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- The nonzero octonions under multiplication
- The sphere S^7 under octonion multiplication
- I have discussed other examples previously:
https://t.co/q5LjmxHEIF
https://t.co/UPHSMwQo75
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Rethinking Vector Addition
— Michael Kinyon (@ProfKinyon) December 1, 2020
or
How I Learned to Stop Worrying and Love Nonassociativity
A thread in 29 tweets
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Now back to the general case where Q is any loop.
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Loops can be studied via their multiplication and inner mapping groups. I will give one example of how this works.
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1. N is the kernel of a homomorphism,
2. N is invariant under Inn(Q),
3. N is a block of Mlt(Q).
Such an N is called a normal subloop of Q. Q is simple if it has no nontrivial normal subloops.
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Theorem (Albert 1941): A loop Q is simple if and only if Mlt(Q) is primitive.
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More from Maths
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BREAKING: @CommonsCMS @DamianCollins just released previously sealed #Six4Three @Facebook documents:
Some random interesting tidbits:
1) Zuck approves shutting down platform API access for Twitter's when Vine is released #competition
2) Facebook engineered ways to access user's call history w/o alerting users:
Team considered access to call history considered 'high PR risk' but 'growth team will charge ahead'. @Facebook created upgrade path to access data w/o subjecting users to Android permissions dialogue.
3) The above also confirms @kashhill and other's suspicion that call history was used to improve PYMK (People You May Know) suggestions and newsfeed rankings.
4) Docs also shed more light into @dseetharaman's story on @Facebook monitoring users' @Onavo VPN activity to determine what competitors to mimic or acquire in 2013.
https://t.co/PwiRIL3v9x
Some random interesting tidbits:
1) Zuck approves shutting down platform API access for Twitter's when Vine is released #competition
![](https://pbs.twimg.com/media/DtqhSTdU4AAJRnB.jpg)
2) Facebook engineered ways to access user's call history w/o alerting users:
Team considered access to call history considered 'high PR risk' but 'growth team will charge ahead'. @Facebook created upgrade path to access data w/o subjecting users to Android permissions dialogue.
![](https://pbs.twimg.com/media/Dtqkp_ZV4AA2o2b.jpg)
3) The above also confirms @kashhill and other's suspicion that call history was used to improve PYMK (People You May Know) suggestions and newsfeed rankings.
4) Docs also shed more light into @dseetharaman's story on @Facebook monitoring users' @Onavo VPN activity to determine what competitors to mimic or acquire in 2013.
https://t.co/PwiRIL3v9x
![](https://pbs.twimg.com/media/Dtqnj5fUUAIyVPc.jpg)