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The direct product of groups with no nonnormal subgroups may have nonnormal subgroups

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.5

Solution: Write $Z_4 = \langle x \rangle$ and consider $\langle (i,x) \rangle = \{ (i,x), (-1,x^2), (-i,x^3), (1,1)\}$.

We have $$(j,1)(i,1)(j,1)^{-1} = (jij^{-1}, xxx^3) = (-i,x);$$ thus $\langle (i,x) \rangle$ is not normal in $Q_8 \times Z_4$.


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