Here are some: Cauchy's integral theorem Cauchy's limit theorem Cauchy's residue theorem Cauchy's mean-value theorem in real analysis Cauchy's theorem on the rigidity of convex polytopes Cauchy-Hardemard theorem in complex analysis Cauchy-Peano theorem for ordinary differential equations Cauchy-Goursat theorem for complex-valued functions Cauchy-Kovalevskaya theorem about partial differential equations
p can divide m too. If so, then |G|=p^2 m' for some m'. So you divide out p as long as it possible, i.e. m is not divisible by p. You get p^k m, for some k which may be 1, but you know that m is not divisible by p.
Sir , What about the lemma ( near about 2.30) if |S| = p? Then |S|(mod p) = 0 so what does it tell? There is no fixed point in S? Or that all the p points of S are fixed? Or what can we say when |S|=p^k?
The "up to isomorphism" part is important. One of the two groups he shows is isomorphic to Z6. Note that Z/2 X Z/3 is cyclic and can be generated by (1,1) for instance. So it is isomorphic to Z/6
I missed where the Lemma is used in the proof of Cauchy's Theorem.
20:40
Why |Orb(s)| can't be grater than p?
Isn't there another Cauchy's theorem from Complex Analysis?
of course, Cauchy is everywhere!
Here are some:
Cauchy's integral theorem
Cauchy's limit theorem
Cauchy's residue theorem
Cauchy's mean-value theorem in real analysis
Cauchy's theorem on the rigidity of convex polytopes
Cauchy-Hardemard theorem in complex analysis
Cauchy-Peano theorem for ordinary differential equations
Cauchy-Goursat theorem for complex-valued functions
Cauchy-Kovalevskaya theorem about partial differential equations
8:18 Where does the `k` comes from? p divides |G|, therefore |G|=pm for some m. Why is there an exponent?
p can divide m too. If so, then |G|=p^2 m' for some m'. So you divide out p as long as it possible, i.e. m is not divisible by p. You get p^k m, for some k which may be 1, but you know that m is not divisible by p.
Sir , What about the lemma ( near about 2.30) if |S| = p?
Then |S|(mod p) = 0 so what does it tell? There is no fixed point in S? Or that all the p points of S are fixed?
Or what can we say when |S|=p^k?
|Fix(phi)|=0 means no fixed points exists.
2:48 p=6?
By assumption, p must be prime
@@dominikchmura5103 I'm not sure, but that might have been a joke on the fact that as was drawn p was 6.
Wait, you say there are only two groups of order 6, but what about Z6?
The "up to isomorphism" part is important. One of the two groups he shows is isomorphic to Z6.
Note that Z/2 X Z/3 is cyclic and can be generated by (1,1) for instance. So it is isomorphic to Z/6