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In $D_4\times\mathbb Z_2$, find normal subgroups $H$ of orders $2$,$4$, and $8$. For each $H$ describe $G/H$.

I know what makes a subgroup normal, but I don't know how to find one. And what would it mean to describe G/H? I could think of is maybe the left cosets and isomorphic subgroups?

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    It's supposed to be D4 x Z22012-12-14

2 Answers 2

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we know that $D_4$ is generated by $a$ and $b$ where $a$ is a permutation of length 4.

  • $H_1= D_4\times \{0\}$ has order 8.

  • $H_2=\{e,a^2b\}\times\{0,2\}$ has order 4.

  • $H_3=\{e\}\times\{0,2\}$has order 2.
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Hints (be sure you can prove the following):

(1) in a direct product, any normal subgroup of one of the factors is normal in the whole group

(2) Besides the above, check the cases where the subgroup is not a subgroup of one of the factors. Sylow subgroups may help here