Exercise 1706Algebra › GroupsL1Added 3 Nov 2026

Normal subgroups

#normal subgroup#conjugation#criterion

Statement

Let $N$ be a subgroup of $G$ . Show that the following are equivalent:
1. $N\triangleleft G$ ;
2. $gNg^{-1}=N$ for all $g\in G$ ;
3. $gNg^{-1}\subseteq N$ for all $g\in G$ ;
4. every left coset is a right coset.

Hint

For (3) $\Rightarrow$ (2), apply (3) to $g^{-1}$ .

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Solution

1 $\Leftrightarrow$ 2. $N$ normal means $gN=Ng$ for all $g$ , i.e. $gNg^{-1}=N$ .

2 $\Leftrightarrow$ 3. If $gNg^{-1}\subseteq N$ for all $g$ , then applying to $g^{-1}$ gives $g^{-1}Ng\subseteq N$ , hence $N\subseteq gNg^{-1}$ , so equality. Conversely, equality implies inclusion.

2 $\Leftrightarrow$ 4. $gN=Ng\iff gNg^{-1}=N$ . The condition that left cosets equal right cosets is the same.

Conclusion. All four are equivalent.

Normal subgroups

Statement

Solution

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