20200722

• Dont giveup keep trying

20200611 (Fri)

Working again

• Dont fill your mind with social media

Hungerford Problem 1.5.19

Suppose \(H\leq G, N\trianglelefteq G\), then \(([G:H],|N|)=1 \rightarrow N < H\) and \(([G:N],|H|)=1) \rightarrow H < N \)

Prove that every permutation is a product of distjoin cycles

• given a cycle, orbit creates a equi relation btw the elements of {1...N}

Cor order of ab is lcm of the order of a and b

Cor order of product of cycles is lcm of the orders of the cycles

• Need to extend this to work on disjoint cycles

Any Sn can be decomposed into transpositions

• \((1234) = (12)(13)(14)\)

20200615 (Tue)

• Show that every permutation can be classified as odd xor even

How can you

• Train your ability to seek out different avenues and attack the problem from these angles?

After working independently and deriving the next two steps of the book, I've come to realizations

• Thinking on your own and asking questions

Even permutations is a normal subgroup

• Subgroup is clear \(a\)

Subgroup is simple if it has no other trivial subgroup

• Odd is not subgroup because it does not contain identity

No other subgroup has index 2

• I thought about how to draw contradiction

Even permutation is the unique subgroup of index 2 in Sn

• I tried many many different approaches

Really difficult and stuck on (prev problem)

• Ok it was a excepttional difficult exercise that requires a hint of use of a lemma

The time is now

• I dont want to be fake. I want to be fing legit

Examples of Sn

• How does S2 extend to S3? or SN to SN+1

20205020

Burn out

• I have stopped making progress on the problem. Why?

210507

HUNGERFORD THM 1.4.8

\(|H:H \cap K| \leq |G:K|\). If \(|G:K|\) is finite \(|H:H \cap K| = |G:K| \leftrightarrow HK = G\)

210506

HUNGERFORD THM.1.4.7

\(|HK| = \dfrac{|H||K|}{|H \cap K|}\)

Break

• Prevent burnout

Warning

• Be mindful how you fill up the pages and what you write. Too much and it will clutter.

Motivation

• If what you are trying to prove is true, there IS a natural solution/connection/reason to get there!

Questions

• Do I understand all the members/elements/components of the problem?