In this talk, we introduce the Hopf algebra HSym of signed permutations. Extracting a permutation from a signed permutation defines a Hopf algebra surjection form HSym to the Malvenuto-Reutenauer Hopf algebra SSym of permutations and taking a suitable weak composition from a signed permutation defines a Hopf algebra surjection from HSym to the Hopf algebra RQSym of weak quasisymmetric functions. Together with the existing Hopf algebra surjections from SSym and RQSym to the Hopf algebra QSym of quasisymmetric functions, we obtain a commutative diagram of Hopf algebras revealing the close relationship among compositions, weak compositions, permutations and signed permutations. This is joint work with Li Guo, Jean-Yves Thibon.

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