Prove that group of order 203 is abelian.

Question

Prove that group of order 203 is abelian.

Answer 1

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.