  Recently, B\'ona and Smith defined strong pattern avoidance, saying that a
permutation $\pi$ strongly avoids a pattern $\tau$ if $\pi$ and $\pi^2$ both
avoid $\tau$. They conjectured that for every positive integer $k$, there is a
permutation in $S_{k^3}$ that strongly avoids $123\cdots (k+1)$. We use the
Robinson--Schensted--Knuth correspondence to settle this conjecture, showing
that the number of such permutations is at least $k^{k^3/2+O(k^3/\log k)}$ and
at most $k^{2k^3+O(k^3/\log k)}$. We enumerate $231$-avoiding permutations of
order $3$, and we give two further enumerative results concerning strong
pattern avoidance. We also consider permutations whose powers all avoid a
pattern $\tau$. Finally, we study subgroups of symmetric groups whose elements
all avoid certain patterns. This leads to several new open problems connecting
the group structures of symmetric groups with pattern avoidance.
