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Ahimsa is from the Sanskrit for non-violence: Origin: Sans ahimsā, non-injury < a-, not + himsā, injury < IE *ĝheis-, to wound < base *ĝhei-, to hurl, projectile.
Hamsa is the Arabic for "five", and since the hand has five fingers (or four + a thumb, for the pedants), it is an obvious connection.
Now, the origin of the hamsa symbol is a whole other question which I have neither time nor space to answer here. Perhaps another time.