Using pumping lemma to prove irregularity of regular language - what is my error?
The pumping lemma gives you a pumping length $n$ and $\alpha, \beta, \gamma$ under the conditions specified in it so that $\alpha \beta^k \gamma \in L$ for all $k \ge 0$. You do not get to choose $\alpha, \beta, \gamma$ as you please. Also, the pumping lemma works for all words of length at least $n$, not simply an arbitrary word $\sigma$.

The trick when using the pumping lemma, then, is picking a word longer than the pumping length and such that any decomposition into $\alpha, \beta, \gamma$ (obeying the conditions specified in the lemma) can be used to derive the contradiction that you need.