In Abstract Algebra, what are some basic properties of left cosets of subgroups of a group? Let G be a group, let H be a subgroup of G, and let "a" be an element of G (written a ∈ G), then: 1) a ∈ aH ("a" is an element of the left coset of H in G containing "a"), 2) aH = H if and only if a ∈ H (the left coset of H in G containing "a" equals H itself if and only if "a" is an element of H), and 3) aH = bH if and only if a^(-1)b ∈ H (two left cosets of H in G, containing "a" and "b", respectively, are equal if and only if a^(-1)b ∈ H.

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