HW7
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HW 7: MAT 312/AMS 351, SPRING 2015 In problems (1)-(4) below questions are asked about permutations σ, π ∈ S 5 defined as follows: σ = 1 2 3 4 5 2 3 1 5 4 ; and π = 1 2 3 4 5 5 1 4 3 2 . (1) Compute σπ and πσ. Do σ and π commute? (2) Compute σ -1 and π -1 . (3) Compute the cycle decomposition for both σ and π. (4) (a) Compute the order of σ and the order of π. (b) Compute σ 99 . (c) Compute π 51 . (5) Let H denote the collection of all the following permutations in S 5 : id, (123)(45), (123), (54), (132), (132)(45). Give the multiplication table for H. (6) Which permutations in problem (5) are even, and which are odd? (7) Write the permutation (127)(35764)(265) (in S 7 ) as a product of tran- positions. (8) Let γ denote a permutation in S n . If sgn(γ )=-1, then show that the order of γ must be odd. 1
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An SBU homework, in mathematics specifically. May be useful to people looking for practice in the necessity, art, establishment of financial situation markers of index.
Transcript of HW7
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HW 7: MAT 312/AMS 351, SPRING 2015
In problems (1)-(4) below questions are asked about permutations
, pi S5 defined as follows: =
(1 2 3 4 52 3 1 5 4
); and pi =
(1 2 3 4 55 1 4 3 2
).
(1) Compute pi and pi. Do and pi commute?
(2) Compute 1 and pi1.
(3) Compute the cycle decomposition for both and pi.
(4)
(a) Compute the order of and the order of pi.(b) Compute 99.(c) Compute pi51.
(5) Let H denote the collection of all the following permutations in S5: id,(123)(45), (123), (54), (132), (132)(45). Give the multiplication table for H.
(6) Which permutations in problem (5) are even, and which are odd?
(7) Write the permutation (127)(35764)(265) (in S7) as a product of tran-positions.
(8) Let denote a permutation in Sn. If sgn()=-1, then show that theorder of must be odd.
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