Suppose that f: G → G such that f(x) = axa. Then f is a group homomorphism if and only if a = e O a^4 = e a^3 = e a^2 = e

Q: Provide an example of the following and explain why it works: 1.) A Galois extension of Q with…

A: Introduction: The Galois group of a certain kind of field extension is a particular group connected…

Q: Let G :- [0, 1) be the set of real numbers x with 0<x< 1. Define an operation + on G by X* y:= {x+y…

A: Here we check associativity property.

Q: Dn Prove that is isomorphic to a subgroup of Sn

A:

Q: Show that SL(n, R) is a normal subgroup of GL(n, R). Further, by apply- ing Fundamental Theorem of…

A: Suppose, ϕ:GLn,R→R\0 such that ϕA=A for all A∈GLn,R Now, sinceA∈GLn,R if and only if A≠0 Now, we see…

Q: 9. Describe the group of the polynomial (x* – 1) e Q[x] over Q.

A:

Q: Show if all primitive transformations of the nonzero form x '= x ,y' = cx + dy d are a group

A: Given:  x '= x ,y' = cx + dy d

Q: 33. If a, bɛR with a # 0, let T:R→ R be the function given by T(x) = ax + b. Prove that the set G =…

A:

Q: Suppose that f (x) is a fifth-degree polynomial that is irreducible overZ2. Prove that x is a…

A:

Q: Without looking at inner automorphisms of Dn, determine the numberof such automorphisms

A:

Q: Let G = {[1 0 0 1] ,[−1 0 0 1] ,[1 0 0 −1], [−1 0 0 −1]} . Is G ∼= K4? If yes, give an explicit…

A:

Q: Ql: Prove that (Q\{0},x) is a subgroup of (R\{0},x).

A:

Q: GX H G, X H. 19. Prove that a group Gis abelian if and only if the function f:G→ G given by f(x) =…

A:

Q: The group GLQ,R) abelian group is an

A:

Q: ii) Show that the function f (x) defined from the group (R, +) to the (R,×) by f (x) = e* is a…

A:

Q: Let G be a group and let g, h ∈ G. Show that | gh | = | hg |. Remember that | a | denotes the order…

A:

Q: 1. Define x*y over R\ {-1}by x*y = x + y +xy. Prove that this structure forms an abelian group.

A:

Q: Let Z[x] be the group of polynomials in x with integer coefficientsunder addition. Prove that the…

A:

Q: There exists a quartic polynomial (i.e. a polynomial of degree 4) over Q whose Galois group is…

A:

Q: Suppose that f: G G such that f(x) = axa*. Then f is a group homomorphism if %3D and only if a = e O…

A:

Q: Let G = (-1,1) and (x,y) --> x ○ y = x + y/1 + xy is a binary operation on G. Let R+ = (0, ∞), with…

A: Given: G,O  with G=(-1, 1) ,  (x, y)→ x O y = x+y1+xy   and ℝ+, ·, ℝ+ = (0, ∞) with standard…

Q: 3. Let f : (R\{0},-) (R\{0}, -) be group homomorphism defined by f(a) = |a|. Then ker(f) %3D =...

A:

Q: Let G={-1,0,1}. Verify whether G forms an Abelian group under addition.

A: G  is a group under '+' if  (i) a , b E G  -----&gt; a + b E G     (ii) a E G called the identity…

Q: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a^2= e a =…

A: Since f is a group homomorphism , where f(x)=a∗x∗a−1,    x∈G. So a^-1=a implies self inverse implies…

Q: %3D Let x belong to a group. If x² +e while x° = e, prove that about the order of r?

A: Given that x2≠e and x6=e To prove that x4≠e and x5≠e Suppose that x4=e  also x6=e therefore…

Q: Use Lutz-Nagell's theorem and reduction mod p theorem to show that the torsion group of E : y² = x³…

A: Given: Use Lutz-Nagell's theorem and reduction mod p theorem. To proof: Torsion group of E:y2=x3+3…

Q: Define a binary operation on R\ {0}by x*y = . 2 Prove that this set with this binary operation is an…

A:

Q: 1. Let 0(V) be the set of all orthogonal transformations on V. Prov O) is a group with respect to…

A: Let O(V) be the set of all orthogonal transformations on V. The determinant of an orthogonal matrix…

Q: let G={h: [0,1] approaches to R: h has an infinite number of derivatives). then G is a group under…

A: We need to find f inverse of (g) and show it is a coset of ker(f).

Q: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a = e O a^4…

A: Given that f:G→G be a function such that fx=axa

Q: Let (G,*) be a group. Show that (G,*) is abelian iff (x * y)² = x² * y² for all x, y E G.

A: If a group G is abelian, then for any two elements x and y,    (x*y) = (y*x)   now associative…

Q: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a^4 = e a^3…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: (2) (Z x Z)/(6Z × 14Z)

A:

Q: (7) Define GL2 (R) to be the group of invertible 2 x 2 matric manifold, cc this group has the…

A: Define GL2R  to be the group of invertible 2×2 matrices.  To prove that this group has the structure…

Q: f:H1 x H2 x ... x Hn ---> H1 + H2 + ... + Hn via h1h2...hn ---> (h1, h2,...,hn

A:

Q: Determine the galois group

A:

Q: If G is a non-abelian group of order 8 with Z(G) {e}, prove that |Z(G)| = 2.

A:

Q: If f: G,→ G, is Select one: а. сyclic b. not abelia C. empty sei

A: Given that, f : G1 to G2 is a group homomorphism. G1 is cyclic. Let e be the identity element of G1.…

Q: Suppose that f:G → G such that f(x) = axa. Then f is a group homomorphism if and only if a^4 = e %3D…

A:

Q: Decide if the abelian group Z/2 × Z/2 is cyclic or not. Prove your answer

A:

Q: Let GL(2,R) be the general linear group of 2 by 2 matrices. 1 x is a) Show that o: R GL (2,R) where…

A: We first prove this is a homomorphism.

Q: Let (G,*) and (H,*) be finite abelian groups. If G x G = H x H then G=H. Show that they are…

A: Given that, G×G=H×H⇒G=H Since G,* and H,* are both finite abelian groups we get,…

Q: show that under complex multiplication, G={1,-1.i,-i} is an abelian group?

A: we have proved this by cayley table.

Q: 15*. Find an explicit epimorphism from Z24 onto a group of order 6. (In your work, identify the…

A: To construct a homomorphism from Z24 , which is onto a group of order 6.

Q: How do you interprete the main theorem of Galois Thoery in terms of subgroup and subfield diagrams?

A:

Q: On Z is defined the following binary operation; x . y = x + y - 1 Show that (Z, . ) is an abelian…

A:

Q: Find the Galois group of the polynomial r-1.

A:

Q: Show if all primitive transformations of the nonzero form x '= x ,y' = cx + dy d are a group.

A:

Q: (4) Find the Galois group of the polynomial r + 1.

A: Since you have asked multiple question, we will solve any one question for you. If you want any…

Q: Suppose now that we have two groups (X,o) and (Y, *). We are familiar with the Cartesian product X x…

A: Let X,◊ and Y,* are two groups. The Cartesian product of X and Y defined by X×Y=x,y | x∈X and  y∈Y.…

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.