Abstract Algebra

Prove #2 using a real-world example.

2. (Verifying Theorem 7.4.) Give an example of a group G with |G| = 0o, and some subgroups H; of G for i E I such that ( H, is neither the iEl empty set o nor the trivial subgroup {e}. Then, verify Theorem 7.4 for Hi. iel7.4 Theorem The intersection of some subgroups H, of a group G for i e I is again a subgroup of G. Proof Let us show closure. Let a E nie/ H; and b E nier Hi, so that a E H; for all i e I and be H; for all i E I. Then ab e H; for all i e I, since H; is a group. Thus ab E nier Hi. Since H; is a subgroup for all i e I, we have e e H; for all i e I, and hence e E niel Hi. Finally, for a E niel Hi, we have a E H; for all i e I, so ace H; for all i e I, which implies that a - E niel Hi. Let G be a group and let a; E G for i e I. There is at least one subgroup of G containing all the elements a; for i e I, namely G is itself. Theorem 7.4 assures us that if we take the intersection of all subgroups of G containing all a; fori e I, we will obtain a subgroup H of G. This subgroup H is the smallest subgroup of G containing all the a; for i e