Prove that group of order 203 is abelian.

  • $\begingroup$ Is 203 prime ?. $\endgroup$
    – Amr
    Nov 13 '16 at 18:50
  • $\begingroup$ 7 x 29 = 203 implies that it has subgroups of order 7 and of order 29 $\endgroup$
    – janmarqz
    Nov 13 '16 at 18:56

This is false, we can build non-abelian groups of order $203$ by taking a semidirect product of $\mathbb Z_7$ and $\mathbb Z_{29}$.

To show one exists you just need to exhibit a non trivial morphism $\mathbb Z_7\rightarrow Aut(\mathbb Z_{29})$.

Recall that $Aut(\mathbb Z_{29})\cong \mathbb Z_{29}^*\cong \mathbb Z_{28}$ ( the last one is because $29$ is prime).

Clearly the homomorphism $f:\mathbb Z_7\rightarrow \mathbb Z_{28}$ defined by $\overline x\mapsto \overline{4x}$ works.

  • $\begingroup$ I fixed the obvious two typos. Hope that's ok. $\endgroup$ Nov 13 '16 at 20:14
  • $\begingroup$ Oh yeah, thank you very much. I don't know what happened there :) $\endgroup$
    – Yorch
    Nov 13 '16 at 21:38

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