If I know that $$g^a \neq 1 \mod b$$ is that always true that if I will take a positive integer $c$ and count $(g^a)^c$, then $$(g^a)^c \neq 1 \mod b$$?
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Hint $\ $ In a finite commutative ring every element is a unit or a zero-divisor. By Lagrange's theorem, a unit has finite order (and the converse is clear), $ $ so the units are precisely the elements of finite order, so the elements you seek, those not of finite order, are precisely the zero-divisors.