It seems misleading to call this "symmetry" because what you wrote is not a group.
The only idempotent element in a group is the identity. And in abelian groups where the binary operation is given by "+" the identity is denoted by "0". Yet you have "1" being idempotent, so conclude that this is not a group (or that this is a sleight of hand because you've made the identification that 1 = 0).
The only idempotent element in a group is the identity. And in abelian groups where the binary operation is given by "+" the identity is denoted by "0". Yet you have "1" being idempotent, so conclude that this is not a group (or that this is a sleight of hand because you've made the identification that 1 = 0).