Prove that for a fixed value of n, the set Un of all nth roots of 1 forms a group with respect to multiplication.

Q: Show that in a group G of odd order, the equation x2 =a has aunique solution for all a in G.

A:

Q: Let m and n be relatively prime positive integers. Prove that the order of the element (1, 1) of the…

A: To prove ℤm×ℤn≅ℤmn ⇔ gcd(m,n)=1 Part 1) Proof of 'only if'  part  ℤm×ℤn≅ℤmn ⇒ℤm×ℤn is cyclic. Let…

Q: . Let N be a finite group and let H be a subgroup of N. If [H| is odd and [N:H] = 2, prove that the…

A:

Q: Label each of the following statements as either true or false   The order of an element of a finite…

A: Theorem :                           In a finite group, the order of a group element…

Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

A:

Q: Prove that there is no simple group of order 300 = 22 . 3 . 52.

A:

Q: 1. Prove that in any group, an element and its inverse have the same order.

A:

Q: Prove A3 is a cyclic group

A: We know that If G be a group of prime order then G is cyclic group

Q: prove that the group G=[a b] with defining set of relations a^3=e, b^7=e, a^-1ba=b^8 , is a cyclic…

A: We need to prove that , group G = a , b  with defining sets of relations a3 = e   ,  b7 = e  also…

Q: If a is a group element, prove that every element in cl(a) has thesame order as a.

A:

Q: e 2πik/n (1 ≤ k ≤ n) is a group under multiplication.

A:

Q: (d) A commutative ring R is a prime ideal of itself. (e) If p and q are primes, then there is a…

A: Solution :

Q: Prove that the trivial G is faithful if and only if G = {1G} representation of a nite group

A:

Q: Let C be a group with |C| = 44. Prove that C must contain an element of order 2.

A:

Q: Let N be a finite group and let H be a subgroup of N. If |H| is odd and [N:H] = 2, prove that the…

A: According to the given information, Let N be a finite group and H be a subgroup of N. H is odd and…

Q: Show that if a group has an even number of elements then there is a element A other than unity such…

A: We have to prove that if a group has an even number of elements then there is aelement A other than…

Q: Suppose G is a finite group of order n and m is relatively prime to n. If g EG and g™ = e, prove…

A:

Q: Prove that there is no simple group of order 280 = 23 .5 . 7.

A:

Q: For every integer n greater than 2, prove that the group U(n2-1)is not cyclic.

A:

Q: Prove that the additive group L is isomorphic to the multiplicative group of nonzero elements in

A:

Q: .) Prove that for every element a of a Group G, Z(G) is a subset of C(a)

A:

Q: Suppose that G is an abelian group with respect to addition, with identity element 0.Define a…

A: The objective is to show that G forms a ring with respect to these operations. Since G is an abelian…

Q: In group theory (abstract algebra), is there a special name given either to the group, or the…

A: Yes, there is a special name given either to the group, or the elements themselves, if x2=e for all…

Q: Prove that in a group, (a-1)-1 = a for all a.

A: By definition (a-1)-1=a are both elements of a-1. Since in a group each element has a unique…

Q: If a, b are elements in an abelian group G, show that (ab)n = anbn  for every positive integer n.

A: It is given that 'a' and 'b' are elements in abelian group G. This means (abn)=(anb).   Let for n=1

Q: If N is a subgroup of an Abelian group, prove that is Abelian. N |

A:

Q: for every element a of a ((a) Prove that group 6, 2 (G) is a subset of

A:

Q: Prove that the trivial representation of a nite group G is faithful if and only if G = {1G}

A: To Prove: The trivial representation of the nite group G is faithful if and only if G =1G

Q: Prove that in a group, (a-1)¯' = a for all a.

A: To prove that in a group (a-1 )-1=a for all a.

Q: Let G be a group with the order of G = pq, where p and q are prime. Prove that every proper subgroup…

A: Consider the provided question, Let G be a group with the order of G = pq, where p and q are prime.…

Q: suppose G is Finite group and FiGgH homogeneity : prove that Ir(6)|iG| Be a

A: Given that G is a finite group and F:G→H is a homogeneity, i.e. F is a homomorphism. To prove  FG |…

Q: Prove that there is no simple group of order 528 = 24 . 3 . 11.

A:

Q: Prove or disprove, as appropriate: If G x H is a cyclic group then G and H are cyclic groups.

A: GIven two groups G and H such that GxH is cyclic. True or false: G and H themselves are cyclic

Q: Given any prime p, we let G be a group of oder 7p⁶. PROVE or DisPROVE that G is not simple

A: Let  G be a group of oder 7p⁶.

Q: Prove that every subgroup of Z is either the trivial group, {0}, or nZ = {nx | x E Z} for some n E…

A: To prove: That every subgroup of ℤ is either the trivial group{0} or nℤ=nxx∈ℤfor some n∈ℕ. Proof:…

Q: Prove that if G is a finite group, then the index of Z(G) cannot be prime.

A:

Q: Q:: (A) Prove that 1. There is no simple group of order 200.

A: A simple group is a nontrivial group whose only normal subgroups are the trivial group and the group…

Q: Show that if p and q are distinct primes, then the group ℤp × ℤq is isomorphic to the cyclic group…

A: We have to show that if p and q are distinct primes, then the group Zp×Zq is isomorphic to the…

Q: Show that if G is a finite group with identity e and with an even number of elements, then there is…

A: Given,If G is a finite group with identity e and with an even number of elements,then there is a ≠e…

Q: Prove that if every non-identity element of a group G is of order 2,then G is abelion

A:

Q: If G is a cyclic group of order n, prove that for every element a in G,an = e.

A:

Q: Let G be a finite group of order 125 with the identity element e and assume that G contains an…

A:

Q: Let G be an infinite cyclic group. Prove that G (Z,+)

A: To show that any infinite cyclic group is isomorphic to the additive group of integers

Q: according to exercise 27 of section 3.1 the nonzero elements of zn form a group g with respect to…

A: We know that a finite group is said to be cyclic if there exists an element of group such that order…

Q: Show that the translations of Rn form a group.

A: we have to show that   the translations of Rn form a group   We know that

Q: Show that each non-zero element of an integral domain , regarded as a member of the additive group…

A:

Q: Prove that in any group, an element and its inverse have the same order.

A: Proof:Let x be a element in a group and x−1 be its inverse.Assume o(x) = m and o(x−1) = n.It is…

Q: Verify that (ℤ, ⨀) is an infinite group, where ℤ is the set of integers and the binary operator ⨀ is…

A:

Q: Let m and n be integers that are greater than 1. (a) If m and n are relatively prime, prove that Zm…

A: