20131119

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Today

What do we need to learn this week.

1. Lagrange's Theorem
   • x(Monday) Introduce the notation $|G|/|H|=[G:H]$. The index of a subgroup.
   • x(Wed) Next Day -> The order of an element divides the order of a group. Any group whose order is prime must be cyclic.
   • x(Friday) Next Day -> If $G$ is a finite group, then $a^{|G|}=e$ for every $a\in G$. Use this to prove Fermat's little theorem, namely that for any integer $a$ we have $a^p\mod p=a\mod p$.
2. Factor groups (We've got everything already, we just need to put it all together)
   • x(Monday) Define Semigroup and show that the set product gives us a semigroup.
   • xDefine Monoid (maybe - maybe not)
   • x (Monday) Show that the set of cosets has an indentity and inverses.
   • x(Monday) Introduce the notation G/H.
   • (Friday, if even) Equivalence Relations (maybe).
3. Introduce External Direct products and the notation $G\times H$ and or $G\oplus H$.
   • x(Monday) Have them do this by drawing Cayley graphs.
   • x(Monday) Define carefully the external product, and then have them prove that its a group.
   • xShow that if $A$ is a subgroup of $G$ and $B$ is a subgroup of $G$, then $A\times B$ is subgroup of $G\times H$.
   • Show that the projection map gives an isomorphism between $G\times \{e}$ and $G$.
   • Show that the product of Abelian groups is still Abelian.
   • When is the product of cyclic groups cyclic?
   • xWhat is the order of an element $(g,h)$? Prove your claim. They can get at the problem by looking at examples.
   • Prove that $G\times H$ is isomorphic to $H\times G$. This just has them review the concept of isomorphic. Should be extremely fast.
4. x(Monday) Look at $U_d(n)$ and the homomorphism that connects them.
   • xUse this homomorphism to show that if $n=pq$ for primes $p$ and $q$, then we just have $\varphi(pq)=(p-1)(q-1)$. They'll need to review the definition of the Euler phi function.
5. First Isomorphism Theorem
   • x(Mon) They've already done it, basically . This should show up immediately after defining factor groups.
   • x(Wed) Prove that if the kernel of a homomorphism is trivial, then the homomorphism is injective. They've already gotten this basically done (with what Tara showed).
   • (Wed) This shows that every normal subgroup is the kernel of some homomorphism. So a subgroup is normal if and only if it is the kernel of some homomorphism.
   • Give reading assignment, see Tara's suggestion.
6. Normal Subgroup Facts
   • x (Mon) If a group is Abelian, then it is a normal subgroup. The center of any group is normal.
   • Is the centralizer of an element a normal subgroup?
   • (Friday) Any subgroup of index 2 must be normal. Give a proof with a picture that connects to identification graphs, and maybe use $xHx^{-1}=H$ is the test for normality.
   • (Wed) Conjugacy classes - Sylow Subgroup introduction.
7. Homomorphism Properties
   • x (Monday-Wed-Friday) Look at the list and start proving more of these facts, 1 by 1.
   • (Wed) We need to work with preimages. Prove that the preimage of a subgroup is a subgroup.
   • (Friday) Prove that the preimage of a Normal Subgroup is normal. (This is an extension of the fact that the kernel is a normal subgroup).
   • (Friday) Is the image of a normal subgroup normal? Give an example.
8. Examples that they've seen in other places
   • Similar matrices. This is an inner automorphism introduction, as well as the beginnings of Sylow p-subgroups.
   • If we do Sylow p-subgroups, then we take care of the key needed for the Fund. Thm. Of Abelian groups. I'll have to explore this.
9. Sylow Subgroups
10. Applications to Other branches of mathematics.
    • Calculus (integration) and cosets.
    • ODEs and the whole coset of kernel solves nonhomogeneous ODEs. It doesn't matter which particular solution you choose, you just need $y=y_p+y+h$.
    • Similarity matrices and conjugacy classes.
    • Linear transformations - homomorphisms.
    • Multivariable Calculus - div, grad, curl, and all that. Short exact sequences.
    • Convolutions and the dirac delta function from Laplace transforms
    • Linear algebra, row reduction an uniqueness of rref
    • Eigenvalue/Eigenvectors problems.
    • Equivalence Classes (topology - quotient spaces)

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