let x be an element of group g. Prove that if |x|=n then x^-1=x^n-1

Q: The elements of the quotient group (/18) are 1- 2- {.-208,-108,008,108,208,.} 3- (0,1,2,3, .8) 4-…

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Q: A group G is cyclic if and only if there exists at G such that G={a^|n  ∈ Z}. True or False then why

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Q: Suppose that g and h induce the same inner automorphism of a group G. Prove that h-'g E Z(G).

A: Given: g , h induce the same inner automorphism of a group G. Prove that: h-1g∈ZG.

Q: Express the group in disparity

A: Topic:- range on number line

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…

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Q: true O false O * -2 2) Let |G| = n and H a subgroup of G then la, + H| = la, + H| = laz * H| = ... =…

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Q: let G=3^n, n is all integers. show that multiplication is an operation on G and show that is an…

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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,…

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

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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: If G is a group then Z(G)◄G. *True False

A: Use the relevant definitions.

Q: 1- The index 2- infinite group 3- permute elements B prove that (cent G,*) is normal subgroup of the…

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

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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: for the cyclic group of order 16 generated by a , show that a^4 generates an invariant subgroup

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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: [S,*]= [Z,+] and [S',*']=[Z,∘] where a∘b=a+b-1. Then ?(n)=n+1 is an isomorphism from [Z,+] to [Z,∘]

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

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Q: Which of the following groups are cyclic? For each cyclic group, list all the generators of the…

A: To identify the given group is cyclic or not.

Q: Let G be a finite semigroup. Prove that there exists e∈ G such that e = e^2.

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Q: Find the index of in the group Z24

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Q: Given a group hG, ∗i and a subset H ⊂ G. Prove that H is a subgroup of G if (i) H 6= ∅, and (ii) if…

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Q: 3) Let G and H be two groups with e and e' as identities. Define P = G x H by P = G x H = { (g,h) |…

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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: Q2 / If a, b and c are elements of a group (G,*) such that c*a=c*b then a=b.

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Q: G is a finite group and f be an automonphism of G satisfying the conditian f(x)= x → x=e for xEG.…

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Q: 2. Is the set Z3 = {0,1,2} form a group with respect to addition modulo 3 how about to…

A: We use caley's table to verify the properties of a group.

Q: Let a and b be elements in a group G. Prove that ab^(n)a^(−1) = (aba^(−1))^n for n ∈ Z.

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Q: What are the eight steps in computing a t-test for dependent groups

A: The paired t is used when the two sample are paired or related samples. When the sample is measured…

Q: Exercise 3: Prove that every element of a finite group is of a finite order.

A: follow next step

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

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Q: Construct the group Z4 building an isomorphism. Z8/ and determine a group that it is isomorphic to…

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Q: Let G be a group of order 25. Prove G is cyclic or g^5=e for all g in G.  Generalize to any group of…

A: The Result to be proved is: If G is a group of order p2, where p is a prime, then either G is cyclic…

Q: Let G be a finite group, prove that there exists m E G such that a ^ m = e for each a E G and where…

A: Let G be a finite group, prove that there exists m E G such that a ^ m = e for each a E G and where…