Let $g_1, \ldots, g_k$ be distinct permutations on a set $\Omega$. Suppose that $G = \langle g_1, \ldots, g_k \rangle$ is an abelian permutation group with only elements of order at most 2. Is it possible that $G = k$? If not, can $G$ be polynomial in $k$?
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$k\leq G\leq 2^k$
. (This might or might not answer the question, depending on what the question actually is.) – Andreas Blass Aug 29 '11 at 1:21