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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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"I really want to break into Product Management"
make products.
"If only someone would tell me how I can get a startup to notice me."
Make Products.
"I guess it's impossible and I'll never break into the industry."
MAKE PRODUCTS.
Courtesy of @edbrisson's wonderful thread on breaking into comics – https://t.co/TgNblNSCBj – here is why the same applies to Product Management, too.
There is no better way of learning the craft of product, or proving your potential to employers, than just doing it.
You do not need anybody's permission. We don't have diplomas, nor doctorates. We can barely agree on a single standard of what a Product Manager is supposed to do.
But – there is at least one blindingly obvious industry consensus – a Product Manager makes Products.
And they don't need to be kept at the exact right temperature, given endless resource, or carefully protected in order to do this.
They find their own way.
make products.
"If only someone would tell me how I can get a startup to notice me."
Make Products.
"I guess it's impossible and I'll never break into the industry."
MAKE PRODUCTS.
Courtesy of @edbrisson's wonderful thread on breaking into comics – https://t.co/TgNblNSCBj – here is why the same applies to Product Management, too.
"I really want to break into comics"
— Ed Brisson (@edbrisson) December 4, 2018
make comics.
"If only someone would tell me how I can get an editor to notice me."
Make Comics.
"I guess it's impossible and I'll never break into the industry."
MAKE COMICS.
There is no better way of learning the craft of product, or proving your potential to employers, than just doing it.
You do not need anybody's permission. We don't have diplomas, nor doctorates. We can barely agree on a single standard of what a Product Manager is supposed to do.
But – there is at least one blindingly obvious industry consensus – a Product Manager makes Products.
And they don't need to be kept at the exact right temperature, given endless resource, or carefully protected in order to do this.
They find their own way.