How do we start this proof? (a) Assume A n is a subgroup of S n. (b)  (c) Assume o(S n ) = n! (d) Nonempty:

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Presentation transcript:

How do we start this proof? (a) Assume A n is a subgroup of S n. (b)  (c) Assume o(S n ) = n! (d) Nonempty:

What is the next step in the proof? (a) Closed: (b) Show A n is a subgroup of S n. (c) Assume o(S n ) = n! (d) Nonempty:

What is the next step in the proof? (a) Let a, b  H(b) Let f, g  A n (c) Let f, g  S n (d) Let a*b  H (e) Let f  g  A n (f) Let f  g  S n

What is the next step in the proof? (a) Identity: (b) Associative: (c) Inverses: (d) Nonempty:

What is the next step in the proof? (a) Let a  H(b) Let f  A n (c) Let f  S n (d) Let a -1  H (e) Let f -1  A n (f) Let f -1  S n

How many elements does S 5 have? (a) 5(b) 10(c) 20(d) 60 (e) 120(f) 200(g) 500(h) 546

How many elements does A 5 have? (a) 5(b) 10(c) 20(d) 60 (e) 120(f) 200(g) 500(h) 546