Statement#

Let $G$ be a group and $H$ be a subgroup of $G$.

Prove that $H$ is a normal subgroup of $G$ (denoted $H \trianglelefteq G$) if and only if $gHg^{-1} = H \quad \text{for all } g \in G$

where $gHg^{-1} = \{ghg^{-1} : h \in H\}$ is the conjugate of $H$ by $g$.

Required Topics#

• Subgroups and cosets
• Normal subgroups
• Conjugation in groups
• Set equality proofs

Key Definitions#

Normal Subgroup: $H \trianglelefteq G$ if $gH = Hg$ for all $g \in G$ (left and right cosets are equal).

Conjugation: The conjugate of $h$ by $g$ is $ghg^{-1}$.

Solution#

Solution coming soon.