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I've wondered why each multiplicative group I tested has modulo n either 1 or -1. Is there a rule?

For example:
$$\mathbb Z_{10}^*=\{1,3,7,9\}\\ 1\cdot3\cdot7\cdot9=-1 \:(mod\:n)$$ or $$\mathbb Z_{15}^*=\{ 1, 2, 4, 7, 8, 11, 13, 14 \}\\ 1\cdot2\cdot4\cdot7\cdot8\cdot11\cdot13\cdot14=1 \:(mod\:n)$$ or $$\mathbb Z_{5}^*=\{1,2,3,4\}\\ 1\cdot2\cdot3\cdot4=-1 \:(mod\:n)$$

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    See also many related posts on [Wilson's Theorem for abelian groups.](http://math.stackexchange.com/search?tab=votes&q=user%3a242%20wilson%20group) – 2017-01-04

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