Learn the definition of a subgroup.
The following video describing the definition of a subgroup along with a few examples.
In this video, the basics of what it means to be a subgroup explain, and some fun problems working with subgroups are shown.
The Subgroup Test
An easier way to show that a subset of a group is a subgroup: just check closure and inverses.
The Center of a Group
The definition of the center of a group is given, along with some examples. Then, a proof that the center of a group is a subgroup of the group is provided.