While studying examples of groups we had groups contained within larger groups. For example, ℤ the group of integers under addition is contained within the larger group ℚ of rationals under addition, which in turn is contained within the group ℝ of reals under addition. The best way to study any algebraic structure is to study its subsets, which themselves have the same structure. Therefore, we study subsets of a group which are groups in their own right. They are called subgroups. Thus we have the following definition.

**Definition 7.1.** *Let G be a group. A subset H of G is called a subgroup of G if H is a group under the operation of G restricted to H.*

**Notation:** If *H* is a subgroup of a group ...

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