# normal is not transitive

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.

Title normal is not transitive NormalIsNotTransitive 2013-03-22 16:00:34 2013-03-22 16:00:34 Wkbj79 (1863) Wkbj79 (1863) 9 Wkbj79 (1863) Definition msc 20A05 msc 12F10 ExampleOfNormalExtension NormalityOfSubgroupsIsNotTransitive