Question

If S is a nonempty subset of a group G , show that is the intersection of the family of all subgroups H such that $S\subseteq H$.

Short answer

It has been proved that is the intersection of the family of all subgroups H such that $S\subseteq H$ .

Step-by-step solution

Restatement

It is a restatement of Theorem 7.18(2).

Let S be a non-empty subset of a group G. Let be the set of all possible products in every order of elements of S and their inverses. Then,

If H is a subgroup of G that contains the set S, then H contains the entire subgroup .

Proof of Theorem

Any subgroup that contains the set S must contain the inverse of every element of S as well.

By closure property of groups, this subgroup must also contain all the possible products in every order of elements of S and their inverses.

Conclusion

Therefore, every subgroup that contains S must also contain the entire group .