Exercise 1708Algebra › GroupsL1Added 3 Nov 2026

Quotient groups

#quotient group#normal subgroup#cosets

Statement

Let $G$ be a group and $N\triangleleft G$ a normal subgroup. Show that the set $G/N$ of left cosets of $N$ forms a group under the operation $(gN)(hN)=ghN$ .

Hint

Well-definedness is the main point: if $gN=g'N$ and $hN=h'N$ , show $ghN=g'h'N$ .

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Solution

Well-definedness. Suppose $g'=gn_1$ , $h'=hn_2$ with $n_1,n_2\in N$ . Then
$g'h'=gn_1hn_2=g(h n_1' h^{-1})hn_2$
where $n_1'=h^{-1}n_1h\in N$ (normality). So $g'h'=ghn_3$ with $n_3\in N$ , hence $g'h'N=ghN$ .

Group axioms.
- Associativity inherited from $G$ .
- Identity: $N$ (since $gN\cdot N=gN$ ).
- Inverse: $(gN)^{-1}=g^{-1}N$ .

Conclusion. $G/N$ is a group of order $|G|/|N|$ (if $G$ finite), called the quotient group.

Quotient groups

Statement

Solution

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