# Lemma: Factor Groups

Let $$( G ,\ast)$$ be a group and let $$N\unlhd G$$ be its normal subgroup. The quotient set $$G/N:=\{aN: a\in G\}$$ of all (left) cosets, together with the binary operation "$$\circ$$":

$(a_1 N)\circ(a_2N):=(a_1\ast a_2)N\quad\quad(a_1,a_2\in G)$

forms a group $$(G/N,\circ )$$, called the factor group of $$G$$ with respect to the normal subgroup $$N$$, (also pronounced as "G modulo N"). In particular:

1. The operation "$$\circ$$" is well-defined, (i.e. it does not depend on the particular choice of representatives $$a_1,a_2$$).
2. The operation "$$\circ$$" is associative.
3. $$(G/N,\circ )$$ is not empty and contains at least the neutral element of $$(G/N,\circ )$$ is $$N$$.
4. The inverse element of an element $$aN$$ of $$(G/N,\circ )$$ is $$a^{-1}N$$.
5. The inverse element of an element $$aN$$ of $$(G/N,\circ )$$ is $$a^{-1}N$$.

Proofs: 1

Proofs: 1 2
Theorems: 3

Github: ### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013