Question

Done Edit Give complete solutions! Problems: (1) Recall, A,, is the alternating subgroup of Sm, i.e., the subgroup consisting of even permutations. Let n > 3. Show that #An= (Hint: this was discussed in class.) (2) Let k <n be natural numbers. Determine the parity (even or odd) of a cyclic permutation in (0102 ...ax) ES (Hint: the answer depends on the parity of k.) Moreet n be a natural number. Determine the parity of the permutation de S Share defined by (0) + for l....... (Hint: the answer depends on :)

Answer 1

Pobode - we have o(sn) = n! Out of n permutations let G, G ., em be even /comutations and O, O, O , O be add permutations. to showo m=k We have mtk = n! - 0 and Sn = {e ... com, O, O, Ox} het te Sn. Let t be an odd permutation Then te-y- _tm-, 70,--,_themet are all members of Sant a - до 21 an 4.. by closure pro 7 Clearly the tree. to ate odd permutatione and are m in naomber But total no. of odd permutation in Sy are K. - MBK — ® L Also, toto .. tok are even bermututting and - are k in number But total no. of even permutation in Sn are m. -aks mu for by- + - k = m. by 0 mtm = ne - 2m = n! m = nb

- 2 مينمط ملط Let x = (1, G₂ - .. Qx) be any cycles of length K. do (Gen) (9 ) --- Car as l _ which is a product of (K-1) 2-cycles. -

4 x . it a l is (kal) lip to acycles if e . product of even no. of an even bermutation is cupon k H odd. . Similarly . - i nie a ť a L i t an ) odd, k i product of odd no. of d-cycles odd permutation it (kt) ť euen.