Alg7.12

From Example Problems
Jump to: navigation, search

Let N\, be any subgroup of the group G\,. The set of left cosets of N\, in G\, form a partition of G\,. Furthermore, \forall u,v,\in G,uN=vN\, if and only if v^{{-1}}u\in N\, and in particular, uN=vN\, if and only if u\, and v\, are representatives of the same coset.

1\in N\, because N\leq G\,

Therefore g=g\cdot 1\in gN\,\,\forall g\in G\, so G=\bigcup _{{g\in G}}gN\,.

To show that distinct left cosets are disjoint, suppose uN\cap vN\neq \emptyset \, and show that uN=vN\,.

Let x\in uN\cap vN\, Then x=un=vm\, for some n,m\in N\,.

Multiply both sides of the second equation by n^{{-1}}.\,

u=vmn^{{-1}}=vm_{1}\, with m_{1}=mn^{{-1}}\in N\,

Now I can write u\, as vM\,, so for some p\in N\,,

up=(vm_{1})p=v(m_{1}p)\in vN\,

This means uN\subseteq vN\,.

Similarly vN\subseteq uN\, and so uN=vN\,. This finishes the first part of the proof.

To show \forall u,v,\in G,uN=vN\, if and only if v^{{-1}}u\in N\,,

uN=vN\implies u\subseteq vN\, which means u=vn\, for some n\in N\,.

u=vn\implies v^{{-1}}u=n\implies v^{{-1}}u\in N\,

Since u\subseteq vN\,, uN=vN\, if and only if u\, and v\, are representatives of the same coset.

Absract Algebra

Main Page