Gauss's lemma
Gauss’s lemma Let be an odd prime, be an integer coprime to . Take the least residues of , i.e. reduce them to integers in . Let be the number of members in this set that are greater than . Then
Euler's Criterion
Euler’s Criterion Let p be an odd prime, and a an integer not divisible by p. Then
Primitive root theorem
Primitive root theorem. Let p be a prime. Then for any d dividing , there are exactly elements of order d in . In particular there are primitive roots mod p.
Fundamental Theorem of Algebra
Fundamental Theorem of Algebra. Every polynomial of degree greater than zero with complex coefficients has at least one zero.
Euler Poincaré formula
The Euler-Poincaré formula describes the relationship of the number of vertices, the number of edges and the number of faces of a manifold. It has been generalized to include potholes and holes that penetrate the solid.
Maximal abelian subgroup ⟺ self Centralizing
The following are equivalent for a subgroup of a group :
- .
- is an abelian subgroup of and (i.e., is a self-centralizing subgroup of ).
- is an abelian subgroup of and it is not contained in any bigger abelian subgroup of .
Alternating group is simple for all n≥5
Let be an integer .
To show that the alternating group is simple, we must show that any normal subgroup of which contains a nonidentity element must be all of .
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