let G=3^n, n is all integers. show that multiplication is an operation on G and show that is an abelian group

Q: Given integer n>0 let G ne the subset of complex numbersof the form e^((2kpii/n)i), k is an element…

A: Note:- A set G is called Group with respect to a binary operation say · if  1. G is closed. 2. G is…

Q: I have an abstract algebra question I'm stuck on:   If G is an abelian group and n is an element of…

A:

Q: Prove that if x is a group element with infinite order, then x^m is not equal to x^n when m is not…

A:

Q: Write under isomorphism all the abelian groups of order 8 and their respective subgroups.

A: To find all the abelian groups of order 8, we can use the fundamental theorem of finitely generated…

Q: Let (A, *) be a group. Let B be a subset of A such that 2|B| > a= b₁ |A|. Show that, for any a in A,…

A:

Q: Let G a finite group, g E G such

A:

Q: If ?^2 = ? for each element “a” of a group G, where “e” is the identity, then prove that G is…

A:

Q: Find all abelian groups (upto isomorphism) of order 3375.

A: We have to find the all abelian groups (upto isomorphism) of order 3375.

Q: An element is called a square if it can be expressed in the form b2for some b. Suppose that G is an…

A: Given: An element is called a square if it can be expressed in the form b2 for some b.G is an…

Q: Let G be a group with identity element e. Prove that for any element a in G, (a^2)^-1 = (a^-1)^2.…

A: We are given that G is a group with an identity element e. In group theory, a branch of mathematics,…

Q: Let G be a group and a ∈ G. Suppose that |a| = 18, find |a^21|. Suppose that |a| = 32, find a…

A: We are given that G is a group, and a ∈ G.A) The order of element a is given as 18. a=18 ⇒a18 = e,…

Q: Computations In Exercises 1 through 6, determine whether the binary operation gives a group…

A: As per the policy we are solving only 3 subparts of the given question. Please repost the question…

Q: Let G be a group and suppose that a * b * c = e. Show that b * c *a = e.

A:

Q: Consider the group G = {x E R such that x 0} under the binary operation x*y=-2xy O x*x*x=4x^3…

A: Multiplication of the elements of the group elements with respect to binary operation

Q: Find any case in which the number of subgroups with an order of 3 can be exactly 4 in the Abelian…

A: Let G be an abelian group of order 108 Find the number of subgroups of order 3. Prove that, in any…

Q: Consider the group G = {x € R such that x* 0} under the binary operation x*y=-

A:

Q: Let X be a group and we then let x and y be an element of X.  Prove that (x*y)^ -1 = a^-1 * b^-1 iff…

A: Since there are some mistakes in given typed question.question may like "Let X be a group and let…

Q: Let G be a group with order 8. If a is an element of G such that the order of a is 4, find the order…

A: The order of an element a in a group G is the smallest positive integer n such that an=e, where e is…

Q: Let G be a group and a ∈ G. The centralizer of  a in G is equal to the centralizer of a^-1 in G.…

A: First let us see the definition of centralizer or normalizer of a in G (definition from I.N.…

Q: each vertex of gra 7. Prove that the order of any subgroup of a finite group divides the order of…

A:

Q: Show that the Galois group of a polynomial of degree n has orderdividing n!

A: Let us consider that the polynomial, fx=∑i=0ncixin, ci∈F Also let K be the splitting field of fx…

Q: Q3\Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.

A:

Q: prove: let g be a group, if g is abelian then (ab)^2 = (a^2)(b^2)

A: Given g is an abelian group.

Q: Let G be a group (not ncesssarily an Abelian group) of order 425. Prove that G must have an element…

A:

Q: Find all abelian groups (upto isomorphism)  of order 67

A: We have to find all abelian groups (upto isomorphism) of order 675.

Q: Let G4 be the set of all functions from R to R of the form f(x) = ax^3, where a is a nonzero real…

A:

Q: Suppose that a is an element of a group G. Prove that if there is some integer n, n notequalto 0,…

A: Suppose that a is an element of a group G. Prove that if there is some integer n, such that n≠0 and…

Q: Show that Every quotient group of abelian grap Gabelian but converse is not tow. I u

A: We have to show that every quotient group of an abelian group G is abelian but converse is not true.

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

A:

Q: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian

A: