Group Theorems: Lagrange, Sylow, Cauchy

1. Lagrange Theorem:
Order of subgroup H divides order of Group G

Converse false:
having h | g does not imply there exists a subgroup H of order h.
Example: Z3 = {0,1,2} is not subgroup of Z6
although o(Z3)= 3 which divides o(Z6)= 6

However,
if h = p (prime number),
=>
2. Cauchy Theorem: if p | g
then G contains an element x (so a subgroup) of order p.
ie.
$latex x^{p} = e $ ∀x∈ G

3. Sylow Theorem :
for p prime,
if p^n | g
=> G has a subgroup H of order p^n:
$latex h= p^{n}$

Conclusion: h | g
Lagrange (h) => Sylow (h=p^n) => Cauchy (h= p, n=1)


Trick to Remember:

g = kh (god =kind holy)
=> h | g
g : order of group G
h : order of subgroup H of G
k : index

Note:
Prime order Group is cyclic
(Z/pZ, +) order p is cyclic & commutative.

Order 4: Z4 not isomorphic to Z2xZ2

Order 6: only Z6 isomorphic Z2xZ3.
Z6 non-commutative

S3 = {1 2 3} ≈ D3 Not Abelian
(1 2)(1 3) = (1 3 2)

(1 3)(1 2) =(1 2 3)

Lagrange: |G|=6
=> order of subgroups in G = 1,2,3,6
6= 2x3
Cauchy : 2|6, 3|6 (2,3 prime)
=> order of elements in G
= 2, 3

Rigorous Calculus: ε-δ Analysis

Rigorous Analysis epsilon-delta (ε-δ)
Cauchy gave epsilon-delta the rigor to Analysis, Weierstrass 'arithmatized' it to become the standard language of modern analysis.

1) Limit was first defined by Cauchy in "Analyse Algébrique" (1821)

2) Cauchy repeatedly used 'Limit' in the book Chapter 3 "Résumé des Leçons sur le Calcul infinitésimal" (1823) for 'derivative' of f as the limit of

$latex \frac{f(x+i)-f(x)}{i}$  when i -> 0

3) He introduced ε-δ in Chapter 7 to prove 'Mean Value Theorem': Denote by (ε , δ) 2 small numbers, such that 0< i ≤ δ , and for all x between (x+i) and x,

f '(x)- ε < $latex \frac{f(x+i)-f(x)}{i}$ < f'(x)+ ε

4) These ε-δ Cauchy's proof method became the standard definition of Limit of Function in Analysis.

5) They are notorious for causing widespread discomfort among future math students. In fact, when it was first introduced by Cauchy in the Ecole Polytechnique Lecture, the French Napoleon top students booed at him and Cauchy received warning from the school.

Note 1: From the textbook 'Calculus' (1980, USA):
"If can't understand the 'ε-δ' definition, just memorize it like a poem - at least better than saying it wrongly."
E.g. “For all ε>0, there exists δ>0, …”

Note 2: George Polya: "The students are not trained in 'ε-δ', teaching them Calculus is like dropping these rules from the sky..."

Newtonian Calculus not rigorous !

Why Newton’s Calculus Not Rigorous?

$latex f(x ) = \frac {x(x^2+ 5)} { x}$ ...[1]

cancel x (≠0)from upper and below => $latex f(x )=x^2 +5 $

$latex \mathop {\lim }\limits_{x \to 0} f(x) =x^2 +5= L=5 $ ...[2]

In [1]: we assume x ≠ 0, so cancel upper & lower x
But In [2]: assume x=0 to get L=5
[1] (x ≠ 0) contradicts with [2] (x = 0)

This is the weakness of Newtonian Calculus, made rigorous later by Cauchy's ε-δ 'Analysis'.