Author: Ow

Continuous function almost everywhere

Definition. A function is called continuous almost everywhere if it is continuous at all points expect at points which constitute a set of measure zero.

Following are two nice examples which illustrate the definition and a possible misinterpretations.

Define a function f:\mathbb{R}\rightarrow \mathbb{R} by

f(x)=1,\quad x\in \mathbb{Q} f(x)=0,\quad x\in \mathbb{Q}^c.

Note that f is discontinuous at all points in \mathbb{R}.

Define a function f:\mathbb{R}\rightarrow \mathbb{R} by

f(x)=\frac{1}{n},\quad x=\frac{m}{n}\in \mathbb{Q} f(x)=0,\quad x\in \mathbb{Q}^c.

Note that f is continuous at all irrational points. It is however discontinuous at all rational points. But, as rational points in \mathbb{R} is a set of measure zero, the function f is continuous almost everywhere.

Sylow theorems

Definition. Let G be a finite group. For any prime p, the subgroup with the highest power of p is called the p-Sylow subgroup of G.

For example. For a group of order 120. The 2-Sylow subgroup has order 8, the 3-Sylow subgroup has order 3, the 5-Sylow subgroup has order 5, and any p-Sylow subgroup with p\neq 2,3,5 is the trivial subgroup.

Theorem 1. Let G be a finite group. For any prime p, there is a p-Sylow subgroup of G. Also, each p-subgroup of G lies in some p-Sylow subgroup of G.

Theorem 2. For any finite group, for a fixed prime p, all p-Sylow subgroups are conjugate to each other.

Theorem 3. For any finite group G of order |G|=p^nm, where prime p does not divide m, let n_p denote the number of p-Sylow subgroups of G. Then n_p\equiv 1 mod p and n_p|m.

Reference- The Sylow Theorems. Keith Conrad.