Normal Subgroup of Group of Order 24/Mistake

Source Work

1996: John F. Humphreys: A Course in Group Theory:

Chapter $12$: Applications of Sylow Theory: $(6)$ Groups of order $24$:

Proposition $12.7$

Mistake

Thus $S_1$ and $S_2$ are both subgroups of $\map {N_G} T$, so $H = \gen {S_1, S_2}$ is a subgroup of $\map {N_T} G$ and hence $T$ is a normal subgroup of $H$.

Correction

The notation:

$\map {N_G} T$

denotes the normalizer of $T$ in $G$: the largest subgroup of $G$ in which $T$ is a normal subgroup.

Hence the notation:

$\map {N_T} G$

makes no sense in this context, and it should be another instance of $\map {N_G} T$.

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $12$: Applications of Sylow Theory: $(6)$ Groups of order $24$: Proposition $12.7$