Definition of the normalizer of a subgroup [duplicate]
Solution 1:
Let $G=GL_2(\mathbb{Q})$, let $N$ be the subgroup of upper triangular matrices with integer entries and 1's on the diagonal, and let $g$ be the diagonal matrix with entries $2,1$. Observe that $gNg^{-1} \leq N$, but $g$ does not normalize $N$. (This is an Exercise on page 88 of Dummit & Foote's Abstract Algebra text.)