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Question Number 193796 Answers: 0 Comments: 0

Let G be a finite group,f be an automorphism of G such that f(x)=x ⇒x=e . Then prove that, (i)∀g∈G, ∃x∈G such that g=x^(−1) f(x). (ii)If ∀x∈G , f(f(x))=x ⇒ G is Abelian.

$${Let}\:{G}\:{be}\:{a}\:{finite}\:{group},{f}\:{be}\:{an}\:{automorphism}\:{of}\:{G} \\ $$$${such}\:{that}\:{f}\left({x}\right)={x}\:\Rightarrow{x}={e}\:. \\ $$$${Then}\:{prove}\:{that}, \\ $$$$\left(\boldsymbol{{i}}\right)\forall{g}\in{G},\:\exists{x}\in{G}\:{such}\:{that}\:{g}={x}^{−\mathrm{1}} {f}\left({x}\right). \\ $$$$\left(\boldsymbol{{ii}}\right){If}\:\forall{x}\in{G}\:,\:{f}\left({f}\left({x}\right)\right)={x}\:\Rightarrow\:{G}\:{is}\:{Abelian}. \\ $$$$ \\ $$

Question Number 193794 Answers: 1 Comments: 0

Let H be a subgroup of (R,+) such that H∩[−1,1] contains a non zero element. Prove that H is cyclic.

$${Let}\:{H}\:{be}\:{a}\:{subgroup}\:{of}\:\left(\mathbb{R},+\right)\:{such}\:{that}\:{H}\cap\left[−\mathrm{1},\mathrm{1}\right]\: \\ $$$${contains}\:{a}\:{non}\:{zero}\:{element}. \\ $$$${Prove}\:{that}\:{H}\:{is}\:{cyclic}. \\ $$

Question Number 193793 Answers: 1 Comments: 4

Question Number 193792 Answers: 1 Comments: 0

Question Number 193790 Answers: 2 Comments: 0

Question Number 193785 Answers: 1 Comments: 0

prove that (1/2)×(3/4)×(5/6)×(7/8)×(9/(10))×...×((99)/(100))<(1/(10))

$${prove}\:{that} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{3}}{\mathrm{4}}×\frac{\mathrm{5}}{\mathrm{6}}×\frac{\mathrm{7}}{\mathrm{8}}×\frac{\mathrm{9}}{\mathrm{10}}×...×\frac{\mathrm{99}}{\mathrm{100}}<\frac{\mathrm{1}}{\mathrm{10}}\:\:\: \\ $$

Question Number 193769 Answers: 0 Comments: 0

Question Number 193768 Answers: 1 Comments: 0

f(x) = 2+∫_0 ^( x) (2t+f(t))^2 dt then ∫_(−1) ^2 f(x) dx =

$$\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{2}+\underset{\mathrm{0}} {\overset{\:\mathrm{x}} {\int}}\left(\mathrm{2t}+\mathrm{f}\left(\mathrm{t}\right)\right)^{\mathrm{2}} \mathrm{dt}\: \\ $$$$\:\:\mathrm{then}\:\underset{−\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:= \\ $$

Question Number 193767 Answers: 1 Comments: 0

prove that c^(log_b a) =a^(log_b c)

$${prove}\:{that}\:{c}^{{log}_{{b}} {a}} ={a}^{{log}_{{b}} {c}} \\ $$

Question Number 193761 Answers: 2 Comments: 0

Question Number 193759 Answers: 2 Comments: 0

Ques. 5 Prove that if a,b are any elements of a group (G, ∗), then the equation y∗a=b has a unique solution in (G, ∗). Ques. 6 a) Show that the set G of all non-zero complex numbers, is a group under multiplication of complex numbers. b) Show that H={a∈G : a_1 ^2 + a_2 ^2 = 1}, where a_1 = Re a and a_2 = Im a is a subgroup of G.

$$\mathrm{Ques}.\:\mathrm{5}\: \\ $$$$\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:\mathrm{a},\mathrm{b}\:\mathrm{are}\:\mathrm{any}\:\mathrm{elements}\:\mathrm{of}\:\:\mathrm{a}\: \\ $$$$\mathrm{group}\:\left(\mathrm{G},\:\ast\right),\:\mathrm{then}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{y}\ast\mathrm{a}=\mathrm{b} \\ $$$$\mathrm{has}\:\mathrm{a}\:\mathrm{unique}\:\mathrm{solution}\:\mathrm{in}\:\left(\mathrm{G},\:\ast\right). \\ $$$$ \\ $$$$\mathrm{Ques}.\:\mathrm{6}\: \\ $$$$\left.\:\:\:\:\:\mathrm{a}\right)\:\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{set}\:\mathrm{G}\:\mathrm{of}\:\mathrm{all}\:\mathrm{non}-\mathrm{zero}\: \\ $$$$\mathrm{complex}\:\mathrm{numbers},\:\mathrm{is}\:\mathrm{a}\:\mathrm{group}\:\mathrm{under} \\ $$$$\mathrm{multiplication}\:\mathrm{of}\:\mathrm{complex}\:\mathrm{numbers}. \\ $$$$ \\ $$$$\left.\:\:\:\:\:\:\mathrm{b}\right)\:\mathrm{Show}\:\mathrm{that}\:\mathrm{H}=\left\{\mathrm{a}\in\mathrm{G}\::\:\mathrm{a}_{\mathrm{1}} ^{\mathrm{2}} \:+\:\mathrm{a}_{\mathrm{2}} ^{\mathrm{2}} \:=\:\mathrm{1}\right\}, \\ $$$$\mathrm{where}\:\mathrm{a}_{\mathrm{1}} \:=\:\mathrm{Re}\:\mathrm{a}\:\mathrm{and}\:\mathrm{a}_{\mathrm{2}} \:=\:\mathrm{Im}\:\mathrm{a}\:\mathrm{is}\:\mathrm{a}\: \\ $$$$\mathrm{subgroup}\:\mathrm{of}\:\mathrm{G}. \\ $$

Question Number 193757 Answers: 0 Comments: 0

Question Number 193756 Answers: 1 Comments: 0

Question Number 193754 Answers: 1 Comments: 0

Question Number 193750 Answers: 1 Comments: 0

Question Number 193749 Answers: 1 Comments: 0

Question Number 193734 Answers: 1 Comments: 0

I = ∫_0 ^∞ ((x^2 (tan^(−1) (x))^2 )/(x^2 −x+2))dx Help!

$$\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{x}^{\mathrm{2}} \left(\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{x}\right)\right)^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{2}}\mathrm{dx} \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$

Question Number 193733 Answers: 1 Comments: 0

Find the ordinary differential equation satisfy by: y = x^n (A + Blogx)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{ordinary}\:\mathrm{differential}\:\mathrm{equation} \\ $$$$\mathrm{satisfy}\:\mathrm{by}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{y}\:\:=\:\:\mathrm{x}^{\mathrm{n}} \left(\mathrm{A}\:\:+\:\:\mathrm{Blogx}\right) \\ $$

Question Number 193726 Answers: 2 Comments: 0

Question Number 193721 Answers: 3 Comments: 0

prove that a^(log_a N) =N

$${prove}\:{that}\:{a}^{{log}_{{a}} {N}} ={N} \\ $$

Question Number 193719 Answers: 1 Comments: 1

Question Number 193715 Answers: 0 Comments: 0

Question Number 193714 Answers: 1 Comments: 3

Question Number 193712 Answers: 0 Comments: 0

Ques. 3 Show that (Z_4 , +) is a group. Hence find the order of the group and of the element 2∈Z_4 , if it exists. Ques. 4 Prove the if a,b are any elements of a group (G, ∗), then the equation y∗a = b has a unique solution in (G, ∗). Help!

$$\mathrm{Ques}.\:\mathrm{3}\: \\ $$$$\:\:\:\:\:\mathrm{Show}\:\mathrm{that}\:\left(\mathbb{Z}_{\mathrm{4}} ,\:+\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{group}.\:\mathrm{Hence} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{order}\:\mathrm{of}\:\mathrm{the}\:\mathrm{group}\:\mathrm{and}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{element}\:\mathrm{2}\in\mathbb{Z}_{\mathrm{4}} ,\:\mathrm{if}\:\mathrm{it}\:\mathrm{exists}. \\ $$$$ \\ $$$$\mathrm{Ques}.\:\mathrm{4} \\ $$$$\:\:\:\:\:\mathrm{Prove}\:\mathrm{the}\:\mathrm{if}\:\mathrm{a},\mathrm{b}\:\mathrm{are}\:\mathrm{any}\:\mathrm{elements}\:\mathrm{of}\:\mathrm{a}\: \\ $$$$\mathrm{group}\:\left(\mathrm{G},\:\ast\right),\:\mathrm{then}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{y}\ast\mathrm{a}\:=\:\mathrm{b} \\ $$$$\mathrm{has}\:\mathrm{a}\:\mathrm{unique}\:\mathrm{solution}\:\mathrm{in}\:\left(\mathrm{G},\:\ast\right). \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$

Question Number 193707 Answers: 1 Comments: 0

Question Number 193706 Answers: 0 Comments: 0

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