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In an arbitrary group, which of the following facts is true about the order of an element $g$ and that of $gH$, where $H$ is a normal subgroup of $G$? Select ALL that apply.

A

If $g$ has finite order in $G$, then $gH$ has finite order in $G/H$.

B

If $gH$ has finite order in $G/H$, then $g$ has finite order in $G$.

C

If $g$ has finite order in $G$, then $gH$ has finite order in $G/H$ and $|g|=|gH|$.

D

If $g$ has finite order in $G$, then $gH$ has finite order in $G/H$ and $|gH|$ divides $|g|$.

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