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MathPrinceps's profile
Laurens Gunnarsen
Laurens Gunnarsen
Laurens Gunnarsen
@MathPrinceps

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Laurens Gunnarsen

@MathPrinceps

Mathematical physicist and mentor to mathematically talented youth. Talent is that which bridges the gap between what can be taught and what must be learned.

Joined June 2012

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    1. John Carlos Baez‏ @johncarlosbaez 27 Aug 2019
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      A group is a set with a way to "add" elements that obeys (x+y)+z = x+(y+z), with an element 0 obeying 0+x=x=x+0, where every element x has an element -x with x + -x = -x + x = 0. Classifying finite groups is really hard. And there are some surprises! (1/n)pic.twitter.com/wvQPiYGpvj

      8 replies 77 retweets 294 likes
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    2. John Carlos Baez‏ @johncarlosbaez 27 Aug 2019
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      Why are there so many groups whose size is a power of 2? You can think of them as funny ways to add strings of bits. The simplest way is to add all the bits separately, "mod 2", as shown below. It's like addition in base 2 but without any carrying! (2/n)pic.twitter.com/HXu9AuBKwM

      2 replies 9 retweets 67 likes
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    3. John Carlos Baez‏ @johncarlosbaez 27 Aug 2019
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      But there are lots of other ways to add bit strings that give groups! For example we can carry as usual when adding in base 2... except for the leftmost entry, where we don't both to carry. (3/n)pic.twitter.com/sMbeYdV54W

      1 reply 2 retweets 26 likes
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    4. John Carlos Baez‏ @johncarlosbaez 27 Aug 2019
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      There are groups where we add bit strings and do our "carrying" in stranger in ways! Below is one of the simpler methods. There are 14 fundamentally different groups with 16 elements. All come from different rules for carrying when we add bit strings! (4/n)pic.twitter.com/GyVcZ8ng6S

      2 replies 2 retweets 27 likes
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    5. John Carlos Baez‏ @johncarlosbaez 27 Aug 2019
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      Here's a cool theorem. Any group whose number of elements is a power of 2 can be gotten from a way of adding bit strings with a weird rule for carrying... where "carrying" only affects the digit directly to the left of the digits you're adding! (5/n)

      5 replies 6 retweets 64 likes
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    6. John Carlos Baez‏ @johncarlosbaez 27 Aug 2019
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      Mathematicians usually say this theorem another way: they say "any finite 2-group is nilpotent". A "finite 2-group" is a group whose number of elements is a power of 2. "Nilpotent" means it can be described using bit strings and carrying in the way I just explained! (6/n)

      3 replies 2 retweets 38 likes
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    7. John Carlos Baez‏ @johncarlosbaez 27 Aug 2019
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      In fact all this stuff about the number 2 works equally well for any prime number p. Any finite p-group is nilpotent... but now, instead of bit strings, we need to describe it strings of integers mod p. Nilpotent groups are an absolute bitch to classify. (7/n)

      2 replies 1 retweet 40 likes
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    8. John Carlos Baez‏ @johncarlosbaez 27 Aug 2019
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      Because 2 is the smallest prime, when you classify groups of size < N, the power of a prime with the highest possible exponent that's < N will always be a power of 2. This is part of why most finite groups are 2-groups! It's "the power of two". 🙃 (8/n, n = 8).

      7 replies 3 retweets 49 likes
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      Laurens Gunnarsen‏ @MathPrinceps 28 Aug 2019
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      Replying to @johncarlosbaez

      Isn't this the place to say: "If I tell you that a natural number N is divisible by 7, with quotient 13, then it follows that N = 91. But if I tell you that a group G has H as one of its normal subgroups, with the quotient group G/H = K, then good luck figuring out what G is!"

      1:05 PM - 28 Aug 2019
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        1. John Carlos Baez‏ @johncarlosbaez 28 Aug 2019
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          Replying to @MathPrinceps

          It's indeed the place to say that! The thread was aimed at people who didn't know what a normal subgroup was. But this is the reason we get so many groups of order 2^10: we need extra data (the "carry" function or "2-cocycle") each time we double the order.

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