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Homework answers / question archive / Prove #2 with the 6th dihedral group example

Prove #2 with the 6th dihedral group example.

2. (Verifying Theorem 7.4.) Give an example of a group G with |G] = 00,

and some subgroups H; of G for i € I such that () H; is neither theies empty set uor the trivial subgroup {ce}. Then, verify Theorem 7.4

for () Hi.

wel

7.4 Theorem = The intersection of some subgroups H, of a group G fori € / is again a subgroup of G.

Proof Let us show closure. Let a € Nye, H; and b € O<,H;, so that a € H; for all i € f and

b € H; foralli € 7. Then ad € H; forall € J, since H; is a group. Thus ab € Nier Hj.

Since H; is a subgroup for all i € 7, we have e € H, for all i € f, and hence

ee Nes Hi.

Finally. for a € Qc, Hj. we have a € Hi, for all i € J. so a7! € H; for allie 7.

which implies that a~! € Nie; Hj. Cf

Let G be a group and let a; € G for i € /. There is at least one subgroup of G containing all the elements a; fori € /, namely G is itself. Theorem 7.4 assures us that

if we take the intersection of all subgroups of G containing all a; for’ € /, we will obtain a subgroup H of G. This subgroup #7 is the smallest subgroup of G containing all the

a forie f.

E D

F c

\A B

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