We give an elementary proof of the following result. Let C be an abelian and irreducible subgroup of the symplectic group Sp(2m, p). Then C is cyclic and embeds in the (multiplicative) subgroup of order pm+1\documentclass[12pt]{minimal}
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\begin{document}$$p^m + 1$$\end{document} of the field of order p2m\documentclass[12pt]{minimal}
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\begin{document}$$p^{2m}$$\end{document}. The proof yields, in fact, a similar result for nonsingular bilinear forms more generally.
