No simple groups of order 66 or 144.

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We look at an "advanced" group theory problem that uses Sylow's Theorems to show that there are no simple groups of order 66 or 144.

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Order 66? Definitely should've waited for tomorrow for Star Wars Day... But an early May the 4th be with you

johnbailey
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A generalization of the first solution is that if you have a group of order m*p^a (as given) where m < p, then it's not simple.
This is because all divisors of m are below p, so you cannot have any divisor of np other than 1 that is congruent to 1 mod p.

f-th
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Awesome. It’s nice to see higher-level math which may not have as wide an audience but is definitely interesting for a certain niche of viewers.

tomatrix
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0:16 Ok, great. It’s always good to try new ideas
28:27 Good Place To Stop

goodplacetostop
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I'd forgotten all of this. It feels good to be reminded.

ImaginaryMdA
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I'm prepping for a grad algebra exam so this will come in handy. Thanks for being so dedicated to your channel, been subscribed since the early days.

technoguyx
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17 minutes in, had to recall that non identity elements can't be in both a Syl-2 and a Sly-3 subgroup because their order must divide 9 and 16. You may :-)

mcmanustony
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Love to see these higher level problems on YouTube. It can be really hard to find accessible resources for petty much anything after calculus.

malchar
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YEAH! this the kinda stuff I was askin for! Well far more so! Im a mathematician turned programmer so I dont get high level proof math stuff to do anymore :(

jermeekable
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Hi Prof Penn. .thanks. .again for sharing your time, energy and knowledge!

jeffreycloete
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at 17:43 the explaination is that if g in a sylowp and sylowq then the order of g divides p and q so it is 1

salim
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12:25: should be “greater than” (which means it cannot be injective)

carl
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Claps for the the second one sir! (|G|=144). Constructing a homomorphism with non trivial kernel is really good idea for proving group is Not simple. Really impressed 🙏

generalmathematics
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This was excellent. Really interesting. Thanks!

eukleidesk
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This is a fantastic video. It clarified many uses of the Sylow Theorems and definitely helped me prep for my exam.

Zagszy
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As someone taking abstract algebra for like the fourth time, this is extraordinarily helpful. Thank you!

WhydYouChangeMyHandle
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Nice! I still remember the homework exercise from Algebra: for each composite number n less than 168 (excluding 60), prove there are no simple groups of order n. And, yeah, 144 is one of the most difficult orders.

samsonblack
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Very excited for upper level maths. As someone with a math degree I've always been disappointed by the dearth of quality high level lectures on the internet.

gnarlybonesful
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I mean, simply use the classification of all finite simple groups :3

matron
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For your case 3, the first subcase can't happen since 16 not cong to 1 mod 9.

Grassmpl