The astute reader might have noticed in our list of examples of groups that some groups are contained in others. E.g. , , the group of symmetries of the icosahedron is contained in *S*_{12} etc. Furthermore, this subset inherits the group operation from the big group.

*Warning : be careful with the cyclic groups **C*_{n}. For example, even though we have set-theoretically (recall that the former is {0, …, 4} while the latter is {0, …, 9}), the group operation is inconsistent. E.g. 3 + 4 = 2 in *C*_{5} but 3 + 4 = 7 in *C*_{10}. Nevertheless, it is possible to “embed” *C*_{5} into *C*_{10}, as we shall see later.

Just as sets have subsets, it is only natural to define the following: a **subgroup** of *G* is a subset *H* which forms a group under the * operation inherited from *G*. To ensure that (*H*, *) forms a group as well, we need to go back to check those 3 axioms of a group. Associativity clearly comes for free, and the only remaining conditions to check are:

- ;
- if , then the product ;
- if , then the inverse .

In fact, it suffices to reduce to the following conditions.

A subset *H* of *G* is a subgroup if and only if *H* is non-empty, and for any , we have .

Clearly the boxed condition is necessary for *H* to be a subgroup. The proof for sufficiency is easy and left as an exercise. Note that non-emptiness of *H* is important, since without it, the empty set would also become a subgroup of *G*, whereas our earlier definition forces *H* to contain *e* at least. This reason may appear cosmetic, but disallowing empty sets as subgroups turns out to be wise in hindsight when we encounter Lagrange’s theorem in future.

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