# Prove that group of order 203 is abelian. [duplicate]

Prove that group of order 203 is abelian.

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

## 1 Answer

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.

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