×

# Left cosets of N in G partition G

Let $$N \leq G$$.

Proposition. The left cosets of $$N$$ in $$G$$ partition $$G$$.

Proof: First we show their union is precisely $$G$$. Since $$N \leq G$$, $$1 \in N$$ so $$g \in gN$$. Hence

$\bigcup_{g \in G} gN = G.$

Next we show that distinct cosets have empty intersection. We argue by contraposition. Assume $$uN \cap vN \neq \emptyset$$. There exist $$m,n \in N$$ such that $$um = vn$$ . So $$m = u^{-1}vn \in N$$, and by closure we have $$u^{-1}v \in N$$ also.

Thus, for all $$x \in N$$, $$u^{-1}vx \in N$$. This implies that $$vx \in uN$$, so $$vN \subseteq uN$$.

A similar argument implies $$uN \subseteq vN$$. Therefore $$uN = vN$$.

Note by Jake Lee
1 year, 3 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$