We show that for closed finite-sized systems with an odd number of real fermionic modes, even in the presence of many-body interactions, there are always at least two fermionic operators that commute with the Hamiltonian. There is a zero mode corresponding to the total Majorana operator, as shown by Akhmerov [Phys. Rev. B 82, 020509 (2010)PRBMDO1098-012110.1103/PhysRevB.82.020509], as well as additional linearly independent zero modes, one of which is (1) continuously connected to the Majorana mode solution in the noninteracting limit, and (2) less prone to decoherence when the system is opened to contact with an infinite bath. We also show that in the idealized situation where there are two or more well-separated zero modes each associated with a finite number of interacting fermions at a localized vortex, these modes have non-Abelian Ising statistics under braiding. Furthermore the algebra of the zero mode operators makes them useful for fermionic quantum computation [S. B. Bravii and A. Y. Kitaev, Ann. Phys. 298, 210 (2002)APNYA60003-491610.1006/aphy.2002.6254].