Using pumping lemma to prove irregularity of regular language - what is my error? I have a vital misunderstanding of the pumping lemma. In the following example I show an example of using it on a regular language to come to incorrect conclusions. What am I doing wrong?

L={ab}, assume the language is regular so by the pumping lemma there exists some n, and σ = αβγ and σ' = αβ^kγ ∈ L for all non negative k.

σ = aaabbb

α = aa

β = ab

γ = bb

then σ'= αβ^2γ for k=2, σ' =aaababbb

σ'∉ L, a contradiction, thus L is not regular.

L as described I know to be a regular language so I would expect to find ∈ L. This is due to my choice of β spanning across two characters but there is nothing I can find in the pumping lemma which forbids this.
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.