The phrase ``normal is not transitive'' can be used as a mnemonic for two statements.

The first is: ``The relation `is a normal subgroup of' is not transitive.'' This means that, if $H \triangleleft N \triangleleft G$ it does not follow that $H \triangleleft G$ See normality of subgroups is not transitive for more details.

The second is: ``The relation `is a normal extension of' is not transitive.'' This means that, if $K/F$ and $L/K$ are normal extensions, it does not follow that $L/F$ is normal. See example of normal extension for more details.