A question about isomorphic of dihedral group

A) Is $D_4 $ is isomorphic to $S_4$?

B) Is $D_{50}$ is isomorphic to $D_5 \times D_5$

C) Is $D_4 \times D_4$ is isomorphic to $C_4 \times C_4 \times C_4$

my attempt: For A) every element of S4 has order of 4 or less, whereas D12 has elements of order 6 and 12. Since they have different orders on their elements, these two groups cannot be isomorphic.

But i don't about B) and C)

  • $\begingroup$ Is question B correct? $\endgroup$ – Learnmore Dec 6 '16 at 15:31
  • $\begingroup$ @learnmore..sorry i am edited now $\endgroup$ – user271336 Dec 6 '16 at 15:32
  • $\begingroup$ A similar line of reasoning to what you have done in part a) will work for b) and c). $\endgroup$ – Mathily Dec 6 '16 at 15:33
  • $\begingroup$ @Mathily..yes , but i cant find the those elements $\endgroup$ – user271336 Dec 6 '16 at 15:35

$D_4$ has eight elements, but $S_4$ has 24.

$D_{50}$ has 100 elements, and so does $D_5 \times D_5$. But $D_{50}$ has an element of order 50; $D_5 \times D_5$ does not.

$D_{4} \times D_4$ has 64 elements and so does $C_4 \times C_4 \times C_4$. But $D_{4} \times D_4$ is nonabelian; $C_4 \times C_4 \times C_4$ is abelian.

  • $\begingroup$ ..i just added my question $\endgroup$ – user271336 Dec 6 '16 at 15:50

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