That last one is not a group.
ℤ/2ℤ is the holiest group, manifesting itself as Gal(ℂ/ℝ), the group of components of GL(n,ℝ), S_n/A_n, and the group of units of ℤ.
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Yes it is. The units of a ring are the invertible elements. Always a group. For ℤ, that's {1, -1} with multiplication.
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