Prove ＜a＞ is a Subgroup of G with Subgroup Test

In Group Theory from Abstract Algebra, if G is a group and a ∈ G, then ＜a＞ is defined to be the set of all integer powers of "a": ＜a＞={a^n | n ∈ ℤ}. This is a subgroup of G, which we prove with the two-step subgroup test. We need show: 0) ＜a＞ is nonempty (this is trivial), 1) ＜a＞ is closed under multiplication, and 2) ＜a＞ is closed under inversion (taking inverses). This is a subgroup test example.    • Abstract Algebra Course, Lecture 1: I...  .

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