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If $p=2$, then your condition on on words of length at most $2^p$ implies that $a$ and $b$ have order two and commute. Therefore they generate a group of order at most four. So, they together cannot act transitively on a large set.
If $p=2$, then your condition on words of length at most $2^p$ implies that $a$ and $b$ have order two and commute. Therefore they generate a group of order at most four. So, they together cannot act transitively on a large set.
If $p=2$, then your condition on words of length at most $2^p$ implies that $a$ and $b$ have order two and commute. Therefore they generate a group of order at most four. So, they together cannot act transitively on a large set.
If $p=2$, then your condition on words of length at most $2^p$ implies that $a$ and $b$ have order two and commute. Therefore they generate a group of order at most four. So, they together cannot act transitively on a large set.
