Cyclic Group Supplement Theorem 1. Let and write n o …
Suppose a, b ∈ hgi. Then a = gk, b = gm and ab = gkgm = gk+m. Hence ab ∈ hgi (note that k + m ∈ Z). Moreover, a−1 = (gk)−1 = g−k and −k ∈ Z, so that a−1 ∈ hgi. Thus, we have checked the three conditions necessary for hgi to be a subgroup of G. Definition 2. If g ∈ G, then the subgroup hgi = {gk: k ∈ Z} is called the ...
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