**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 (**g**od =**k**ind **h**oly)

=> h | g

g : order of group G

h : order of subgroup H…

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