Sometimes, part of the group is also a group. In that case, the part is called a subgroup. For example, the even numbers form a subgroup of the integers, when the operation is addition, because adding any two even numbers gives another even number. Another example is the group of isometries of a shape, which is the set of ways the shape can be moved, including rotating or mirroring it. Performing any two movements in a row gives another movement, so the set of movements forms a group. A subgroup of movements that do not rotate the shape, but only shift it to another place in space, form a subgroup.
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