# (2) (20 points) (a) Prove that if H < G has index 2 the H is...

(2) (20 points)
(a) Prove that if H < G has index 2 the H is normal.

(b) Using the table for  at the end of the exam, find all normal subgroups of

(c) Find the commutator subgroup

TABLE 1. Table for V. * ро Pi P2 P3 Po Po P1 Mi M2 81 82 P2 P3 M1 M2 81 82 P1|p2|p3|po|di|d2|M2|M P2 P2 P3 PO P1 M2 M1 82 81 P3 |p3|PO|PI|P2 82 di M1 M2 pi M2 M2 di Mid2|p2|PO|PI|P3 di di M1 82 M2 P1 P2 PO P2 82 82 M2 81 MP3 P1 P2 Po Subgroups of D4 {po, P2, M1, M2}, {Po, P2, P1P3}, {P0, P2, 81,82} {P0, P2}, {Po,, M1}, {Po, M2}, {Po,81},{P0,82} {po}

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