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For each of the following statements, decide if it is TRUE or FALSE.
Let $p$ be a prime number. If $H$ is a non-normal subgroup of a group of order $2p$, then $H$ has order 2.
There exists a group $G$ with trivial center that has a normal subgroup of order 2.
If $H$ is a subgroup of $G$ of index two, then $H$ contains every element of odd order in $G$.