Order of an element of a group I have the following example given:

Example:
  The order of 6 in $\langle \mathbb Z_{20}; \oplus, \ominus, 0\rangle$ is
  10. This can be seen easily since $60=10\cdot 6$ is the least common multiple of 6 and 20. The order of 10 is 2, and indeed 10 is
  self-inverse.

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Def 1: Let $G$ be a group and let $a$ be an element of $G$. The order
  of a, denoted ord(a), is the least $m\geq 1$ such that $a^m=e$, if
  such an $m$ exists, and $ord(a)=\infty$ otherwise.

Note that $\langle \mathbb Z_m;\oplus\rangle$ denoted the integeres modulo $m$ with addition modulo $m$ as the only binary operation. (same for multiplication)
Question 1: Does $\ominus$ define the unary operation of "taking" the negative?
Question 2: Can a group have several operations defined on it? If so, can they have the same arity?
Question 3: If Question 2 is true, does it mean, that a given element $a$ has a order for each operation?
 A: Q1: There is a natural isomorphism between a group and its opposite group given by the map $x \rightarrow -x$ using additive notation or $x \rightarrow x^{-1}$ using multiplicative notation so every group admit a meaningful unitary operator of this kind. I prefer to think of them as "taking inverses" rather than "taking negatives" since for a general group additive notation is not often used.
Q2:A group cannot have several operations defined on it because part of the definition of the group is the operation. However a set may have different operations defined on it which each give a different group structure. For example the set of four elements has two different group structures depending on the binary operation.
Q3: The order of an element in a group depends on the binary operation. For example in the cyclic group of four elements there exists an element of order 4, but in the Klein-4 group there are no such elements. You can also change the operation and set to get the same group up to isomorphism. For example consider the n-th roots of unity which form a group under complex multiplication that is isomorphic to the integers mod n. So even though the sets and operation have changed the structure of the group as an algebraic structure is essentially identical.
