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Abstract Algebra

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Moderate

Normality in Groups of Order 8

ABSALG-F7YVL1

There are five groups of order 8 up to isomorphism. Identify which of these groups contain subgroups which are not normal.

Select ALL that apply.

A

$\mathbb{Z}/8\mathbb{Z}$

B

$\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$

C

$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$

D

Dihedral group of order 8: $D_{8}=\langle r,s: r^4=s^2=(rs)^2=1\rangle$ (also commonly denoted $D_4$)

E

The quaternion group: $Q_8=\{\pm 1, \pm i, \pm j, \pm k\}$ with multiplication given by $i^2=j^2=k^2=ijk=-1$ and usual rules for multiplying by $\pm 1$.

F

None of the above.