Groups of Order 30/C 3 x D 5/Mistake
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Source Work
1996: John F. Humphreys: A Course in Group Theory:
- Chapter $13$: Direct products:
- Exercise $2$ (solution)
Mistake
- Now let $N$ be the subgroup of $G$ generated by $x^3$ and $y$. Note that since
- $y x^3 y^{-1} = \paren {y x y^{-1} }^3 = \paren {x^4}^3 = x^{12} = x^{-3}$,
- $N$ is isomorphic to the dihedral group $\map D 3$. ...
- ... To show that $x$ normalises $N$, note that
- $x x^3 x^{-1} = x^3$; and $x y x^{-1} = y x^4 x^{-1} = x y^3 \in N$.
Correction
There are two mistakes here:
- $(1): \quad$ In the first section, it should say:
- $N$ is isomorphic to the dihedral group $\map D 5$.
- $(2): \quad$ That second expression should read:
- $x y x^{-1} = y x^4 x^{-1} = y x^3 \in N$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $13$: Direct products: Exercise $2$