Question and Answers Forum

1) Compute in S_a , a^(−1) ba where a=(1 2)(1 3 5), b=(1 5 7 1) 2) Given permutation α = (1 2)(3 4), β = (1 3)(5 6). Find a permutation x∈S_6 ∃αx = β. help!

$$\left.\mathrm{1}\right)\:\mathrm{Compute}\:\mathrm{in}\:\mathrm{S}_{\mathrm{a}} \:,\:\mathrm{a}^{−\mathrm{1}} \mathrm{ba}\:\:\mathrm{where}\: \\ $$$$\mathrm{a}=\left(\mathrm{1}\:\mathrm{2}\right)\left(\mathrm{1}\:\mathrm{3}\:\mathrm{5}\right),\:\mathrm{b}=\left(\mathrm{1}\:\mathrm{5}\:\mathrm{7}\:\mathrm{1}\right) \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\mathrm{Given}\:\mathrm{permutation}\:\alpha\:=\:\left(\mathrm{1}\:\mathrm{2}\right)\left(\mathrm{3}\:\mathrm{4}\right), \\ $$$$\beta\:=\:\left(\mathrm{1}\:\mathrm{3}\right)\left(\mathrm{5}\:\mathrm{6}\right).\:\mathrm{Find}\:\mathrm{a}\:\mathrm{permutation} \\ $$$$\mathrm{x}\in\mathrm{S}_{\mathrm{6}} \:\exists\alpha\mathrm{x}\:=\:\beta. \\ $$$$ \\ $$$$\mathrm{help}! \\ $$

Question Number 192341 Answers: 1 Comments: 0

1) Find the sign of odd or even (or pality) of permutation θ=(1 2 3 4 5 6 7 8) 2) prove that any permutation θ:S→S where S is a finite set can be written as a product of disjoint cycle help!

$$\left.\mathrm{1}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{sign}\:\mathrm{of}\:\mathrm{odd}\:\mathrm{or}\:\mathrm{even}\:\left(\mathrm{or}\:\mathrm{pality}\right) \\ $$$$\mathrm{of}\:\mathrm{permutation}\:\theta=\left(\mathrm{1}\:\mathrm{2}\:\mathrm{3}\:\mathrm{4}\:\mathrm{5}\:\mathrm{6}\:\mathrm{7}\:\mathrm{8}\right) \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\mathrm{prove}\:\mathrm{that}\:\mathrm{any}\:\mathrm{permutation} \\ $$$$\theta:\mathrm{S}\rightarrow\mathrm{S}\:\mathrm{where}\:\mathrm{S}\:\mathrm{is}\:\mathrm{a}\:\mathrm{finite}\:\mathrm{set}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{written}\:\mathrm{as}\:\mathrm{a}\:\mathrm{product}\:\mathrm{of}\:\mathrm{disjoint} \\ $$$$\mathrm{cycle} \\ $$$$ \\ $$$$\mathrm{help}! \\ $$

Question Number 192340 Answers: 1 Comments: 0

Prove that the order of any permuta− tion θ is the least common multiple of the length of its disjoint cycles. hi

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{order}\:\mathrm{of}\:\mathrm{any}\:\mathrm{permuta}− \\ $$$$\mathrm{tion}\:\theta\:\mathrm{is}\:\mathrm{the}\:\mathrm{least}\:\mathrm{common}\:\mathrm{multiple} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{its}\:\mathrm{disjoint}\:\mathrm{cycles}. \\ $$$$ \\ $$$$\:\mathrm{hi} \\ $$

Question Number 192339 Answers: 1 Comments: 0

Express as the product of disjoint cycle the permutation a) θ(1)=4 θ(2)=6 θ(1)=5 θ(4)=1 θ(5)=3 θ(6)=2 b) (1 6 3)(1 3 5 7)(6 7)(1 2 3 4 5) c) (1 2 3 4 5)(6 7)(1 3 5 7) Find the order of each of them help!

$$\mathrm{Express}\:\mathrm{as}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{disjoint}\: \\ $$$$\mathrm{cycle}\:\mathrm{the}\:\mathrm{permutation} \\ $$$$\left.\mathrm{a}\right)\:\theta\left(\mathrm{1}\right)=\mathrm{4}\:\:\theta\left(\mathrm{2}\right)=\mathrm{6}\:\:\theta\left(\mathrm{1}\right)=\mathrm{5}\:\:\theta\left(\mathrm{4}\right)=\mathrm{1} \\ $$$$\theta\left(\mathrm{5}\right)=\mathrm{3}\:\:\theta\left(\mathrm{6}\right)=\mathrm{2} \\ $$$$ \\ $$$$\left.\mathrm{b}\right)\:\left(\mathrm{1}\:\mathrm{6}\:\mathrm{3}\right)\left(\mathrm{1}\:\mathrm{3}\:\mathrm{5}\:\mathrm{7}\right)\left(\mathrm{6}\:\mathrm{7}\right)\left(\mathrm{1}\:\mathrm{2}\:\mathrm{3}\:\mathrm{4}\:\mathrm{5}\right) \\ $$$$ \\ $$$$\left.\mathrm{c}\right)\:\left(\mathrm{1}\:\mathrm{2}\:\mathrm{3}\:\mathrm{4}\:\mathrm{5}\right)\left(\mathrm{6}\:\mathrm{7}\right)\left(\mathrm{1}\:\mathrm{3}\:\mathrm{5}\:\mathrm{7}\right) \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{order}\:\mathrm{of}\:\mathrm{each}\:\mathrm{of}\:\mathrm{them} \\ $$$$ \\ $$$$\mathrm{help}! \\ $$

Question Number 192338 Answers: 2 Comments: 0

y= ((((lim_(h→0) (((x+h)^3 −x^3 )/h))(Σ_(n=0) ^∞ (x^(n+1) /(n+1))))/(∫_0 ^( x) lnt dt))) (dy/dx)?

$${y}=\:\left(\frac{\left(\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left({x}+{h}\right)^{\mathrm{3}} −{x}^{\mathrm{3}} }{{h}}\right)\left(\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{x}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}\right)}{\int_{\mathrm{0}} ^{\:{x}} {lnt}\:{dt}}\right) \\ $$$$ \\ $$$$\frac{{dy}}{{dx}}? \\ $$

Question Number 192336 Answers: 1 Comments: 0