그룹의 오더(order)와 원소의 오더

The Order of a group G is the number of elements in G, which is denoted by |G|.

The Order of an element a ∈G is the number n such that aⁿ = e in G.
it's denoted by |a| ,sometimes o(a).

For example, Z₃= { 0 , 1, 2 }

the order of 0 = 1 , |0| = o(0) = 1
the order of 1 = 2 , |1| = o(1) = 2
the order of 2 = 2 , |2| = o(2) = 2

Consider about a cyclic subgroup <a> of G with aⁿ = e.

then <a> = { 1, a¹ , a² , … , a^(n-1) }

therefore, order of a cyclic subgroup generated by a is the order of a

that is, " o(a) = |a| = |<a>| "