Problem 5.1.5 (Nonnormal subgroup of $Q_8 imes Z_4$)

Question

Exhibit a nonnormal subgroup of $Q_8 	imes Z_4$ (note that every subgroup of each factor is normal).

Richard Ganaye · Accepted Answer

\begin{proof} 
Consider the subgroup $H$ of $Q_8 	imes Z_4$, where $Z_4 = \langle x angle$ and $|x| = 4$, defined by
$$H = \langle (i, x) angle = \{(1,1) ,(i,x) , (-1,x^2) , (-i, x^3)\}.$$
Then $(i,x) \in H$, but
$$(j,1)(i,x) (j,1)^{-1} = (jij^{-1}, x) = (-i, x) 
ot \in H.$$
So $H$ is not normal in $Q_8 	imes Z_4$.
\end{proof}

Problem 5.1.5 (Nonnormal subgroup of $Q_8 \times Z_4$)

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Problem 5.1.5 (Nonnormal subgroup of $Q_8 \times Z_4$)

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