Abstract
ABSTRACf. We study the symmetricity of the Whitehead element Wn E 1t2np_ 3(S2n-l) for an odd prime p. It is shown that Wo considered as a map S20p-3 -+ S20-1 factors through the p-fold covering map (]: S20p-3 -+ L 20 p-3 only when n is a power of p, and that wp' actually factors through (] if 0 :::;; i :::;; 4. This is some of an odd prime version of the results of Randall and Lin for the projectivity of the Whitehead product ['20-1, '20-1] E 1t40_3(S20-1).
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