(a) G = Z where a * b = a + b – 1 for all a, b EG (b) G = {r € Q |r > 0 } where a * b = ab for all a, b e G (c) G = R\{1} where a * b = a – ab + b for all a, b e G

Q: Write the Caley table of all abelian groups of order 8

A: We have to write cayley table of abelian group of order 8.

Q: Prove that any cyclic group is abelian.

A:

Q: Prove the next statement: If G is an abelian group of finite order n, then G has a subgroup of…

A: Answer

Q: Prove O3 is not a group

A:

Q: Let m ≥ 3. Prove that the symmetric group on m letters is not abelian.

A: We have to show that the permutation of m letters, m≥3 that is Sm is not abelian. For proving the…

Q: Show if the given set is a group defined by the given operation. Check all properties. If so, check…

A:

Q: For any two elements a and b of a group G, show that G  is abelian iff…

A:

Q: prove that a group of order 45 is abelian.

A:

Q: Let a be a group element of even order. Prove that a2 is not in cl(a).

A: Let G be a group and a is an element of even order in the group. Prove that a2 is not in Cla.…

Q: Prove that if G is a group with the property that the square of everyelement is the identity, then G…

A:

Q: Let G be an Abelian group. Determine all homomorphisms fromS3 to G.

A: Given: G is an Abelian group.To find: All homomorphisms from S3→G.

Q: Prove that Z ⨁ Z is not cyclic. Does your proof work for Z ⨁ Gwhere G is any group with more than…

A: To Do: Z⊕Z is not cyclic. Does the proof work for Z⊕G where G is any group with more than one…

Q: Find the non-isomorphic abelian groups of  order 360.

A: We have to find the non -isomorphic abelian group of order 360.

Q: Let G be an Abelian group of order 16. Suppose that there are elementsa and b in G such that |a| =…

A: Given: G be an Abelian group of order 16 and a=b=4.To find: The isomorphism class of G.

Q: ose a Let X be a finite set with at least two elements. Suppose group Gr acti transitively on X.…

A: Please find the answer in next step

Q: Show that a group of order pn, where p is prime, is solvable.

A: Let G be a group of order p^n, where p is a prime number.   We will prove by induction on n that G…

Q: Assume that G is an abelian group. Let H C G be a subgroup. Prove that the factor group G/H is…

A:

Q: 1. Let (G, +) and (H,*) be groups. The direct product of G and H is the set G × H equipped with…

A:

Q: For any group G then G/Z(G) is abelin Select one: O True O False

A:

Q: lternating group An is abelian if and only if n <=3.

A:

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

A:

Q: Let a be a group element of even order. Prove that a2 is not in cl(a).

A: Given that a be a group element of even order. We have to prove that a2 is not in cla.

Q: Prove the statement: Let g be a group, if g is abelian then (ab)2 =  a2b2.

A:

Q: Every cyclic group is necessarily abelian but the converse is not necessarily true

A: Given statement :Every cyclic group is necessarily abelian but the converse is not necessarily true…

Q: Let S = R\{-1} and define a binary operation on S by a * b = a +b+ ab. Prove that S is an abelian…

A: To show that S is an abelian group, we have to prove all these properties 1) S is closed under…

Q: Prove that An even permutation is group w.y.t compostin Compostin function.

A:

Q: Every cyclic group is necessarily abelian but the  converse is not necessarily true.

A: Given statement :Every cyclic group is necessarily abelian but the converse is not necessarily true…

Q: Let G be a group, and a, b € G. Prove that b commutes with a if and only if b- commutes with a.

A:

Q: Exercise 14. Let G₁ and G₂ be a groups. Then the set G₁ X G₂ is a group with the following binary…

A:

Q: Let X be any non-empty set. Show that the set of all bijections of X is a group under the operation…

A: Given: X is any non-empty set.To show: The set of all bijections of X is a group under the operation…

Q: Every cyclic group is necessarily abelian but the converse is not necessarily true

A: .

Q: Prove that a group of order 45 is abelian.

A: We have to proof that a group of order 45 is abelian.

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

A: Given: ℝn. We have to show that: The translations of ℝn form a group. We know that the orthogonal…

Q: 6. Let m > 3. Prove that the symmetric group on m letters is not abelian.

A:

Q: Show that if G and H are isomorphic group, then G commutative implies H is commutative also.

A:

Q: Q)Let G be a group such thatx=x- for each xeG. Show that G is Abelian

A: Given : G is a group such that x=x-1 for each x∈G   To prove : G is abelian.

Q: Let G be a group. Prove that (ab)1= a"b-1 for all a and b in G if and only if G is abelian

A: First, consider that the group is abelian. So here first compute (ab)-1 for a and b belongs to G,…