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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## **closed** as off topic by Alex Bartel, Andreas Blass, Felipe Voloch, Gerry Myerson, S. Carnahan♦ Aug 29 '11 at 2:11

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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