LaTeX inconsistent spacing? Is this normal?

For very mysterious reasons, latexsym defines \unlhd as a \mathbin instead of a \mathrel.

On the other hand, amssymb provides a better symbol, \trianglelefteq.

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}

\renewcommand{\unlhd}{\trianglelefteq}

\begin{document}

\subsection{Third Isomorphism Theorem}
Let $G$ be a group, and $N\unlhd G$, $K\unlhd G$ such that $N\subseteq K\subseteq G$. Then $(G/N)/(K/N)\cong G/K$.
\subsection{Correspondence Theorem}
Let $N\unlhd G$. There exists a bijection $\phi:\{\text{all subgroups $H$ such that $N\subseteq H\subseteq G$}\}\to\{\text{subgroups of $G/N$}\}$, with $\phi(H)=H/N$.

\end{document}

the difference here is caused by the different treatment of two different classes of symbols when the lines in which they occur are justified.

the two classes involved here are \mathbin (for \unlhd, a binary operator) and \mathrel (for \subseteq, a relation).

in the second example, the line is too long to fit, so spacing is compressed where permitted. compression is not applied as severely around relations, but it is applied around binary operators. in the first example, there is no place to break in the math string at the end of the sentence, so all the spaces in the first line are exaggerated, making the difference more obvious.

it's not really a good idea to try to "correct" this manually. a better approach is to reword the two examples so that they don't encounter the problem. (another approach is to use ragged right, so that all spaces are set at their "natural" width.)