The question is from Herstein. Until now, semi direct product has not been covered in the material. I am solving these type of questions using the procedure given by Herstein in previous sections. I was able to solve it for order 55 but for order 203, I am facing an issue. The procedure goes like this ::
Required Order = 203 = 29 * 7
Consider a cyclic group G of order 29.
G = $<a>$ such that $a^{29} = e$
And consider an automorphic mapping $\phi$: $a^i \to a^{6i}$. Ideally order of $\phi$ should be 7 here. i.e. $\phi ^7 = I$.
But In this case, $\phi ^7 = a^{279936i} = a^{-i}$.
If $\phi ^7=I$ was true in this case, then next steps would be to consider a group containing all formal elements $x^ia^j$. This group is the required group asked for in the question which is of order 203.
where i = 0,1,2,3,4,5,6. and j = 0,1,2, ...... ,28.
And symbol x is subject to few conditions which are ::
$x^7=e$
$x^{-1}a^ix = a^{6i}$
$x^ia^j = x^ka^l$ iff i $\equiv$ k (mod 7) and j $\equiv$ l (mod 29).
Please let me know where I am going wrong, why $\phi^7$ isnot equal to I ?