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Prove/Disprove:

$D_{12}$ is isomorphic to $\Bbb Z_3\oplus D_4.$

$D_{12}$ has 11 elements of order $12$ whereas $\Bbb Z_3\oplus D_4$ has 9 elements of order $12$ since $\Bbb Z_3$ has $3$ elements of order $3$ and $D_4$ has $3$ elements of order $4$.

So they are not isomorphic.

Is this true?

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    $\begingroup$ Why "sub"group in your title ? $\endgroup$ – Jean Marie Sep 22 '17 at 9:58
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    $\begingroup$ Your counting is very bad! They both have $4$ elements of order $12$, so that does not work, but they have different numbers of elements of order $2$. $\endgroup$ – Derek Holt Sep 22 '17 at 14:01
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I think you want to check whether $D_{12}$ is isomorphic to $\mathbb{Z}_3 \oplus D_4$ or not. The answer is no. Explanation:

Each group has $4$ elements of order $12$, $2$ elements of order $4$ and $2$ elements of order $3$. But $D_{12}$ has only $2$ elements of order $6$ whereas $\mathbb{Z}_3 \oplus D_4$ has $10$ elements of order $6$.

Further, $D_{12}$ has $13$ elements of order $2$ while $\mathbb{Z}_3 \oplus D_4$ has $5$ elements of order $2$.

You can verify this by counting number of rotations & reflections in $D_{12}$, and by taking l.c.m of orders in $\mathbb{Z}_3 \oplus D_4$.

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  • $\begingroup$ how does $D_{12}$ have 4 elements of order $12$ $\endgroup$ – Learnmore Sep 22 '17 at 11:32
  • $\begingroup$ $D_{12}$ has a subgroup isomorphic to $C_12$, the cyclic group of order 12. This subgroup has $4$ generators. $\endgroup$ – David Hill Sep 22 '17 at 15:47
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I think you want to check whether $D_{12}$ is isomorphic to $\mathbb{Z}_3 \oplus D_4$ or not. The answer is no. Explanation:

Each group has $4$ elements of order $12$, $2$ elements of order $4$ and $2$ elements of order $3$. But $D_{12}$ has only $2$ elements of order $6$ whereas $\mathbb{Z}_3 \oplus D_4$ has $10$ elements of order $6$.

Further, $D_{12}$ has $13$ elements of order $2$ while $\mathbb{Z}_3 \oplus D_4$ has $5$ elements of order $2$.

You can verify this by counting the number of rotations & reflections in $D_{12}$, and by taking l.c.m of orders in $\mathbb{Z}_3 \oplus D_4$.

$D_{12}$ has a unique cyclic subgroup of order $12$ which is generated by rotation. Every cyclic group of order $n$ has $\phi(n)$ generators. Here $n =12$ and $\phi(12)=4$. To better understand $D_n$, try $D_3$ or $D_4$ or both.

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  • $\begingroup$ You have posted two duplicate answers. $\endgroup$ – lhf Sep 22 '17 at 13:14

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