Prove that a group G is abelian if and only if (ab) = a¬!b-1 for all a and b in G.

Q: 1. Let G be an abelian group with the identity element e. If H = {x²|x € G} and K = {x € G|x² = e},…

A:

Q: (G, .) a group such that a.a = e for all a EG.Show that G is an abelian group. Let be

A:

Q: Let R = R \ {-1} and define the operation ♡ on R by a♡b = ab + a + b Va, b E R. Show that (a) ♡ is a…

A:

Q: Let G be a group such that a^2 = e for each a e G. Then G is * О Сyclic O None of these O…

A:

Q: Prove that a group of order 175 is Abelian.

A: Let G be a group of order 175 We first try to rewrite 175 as prime factorization as follows: 175 =…

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

A: (G, *) be a finite group of prime order   To prove (G, *) is an abelian group

Q: (c) Prove that if G is a (not necessarily abelian) group, a, b e G, and a² = b² = (ab)² = e, then ab…

A: Use property of group and solve it.

Q: If G is a finite group and some element of G has order equal to the size of G, we can say that G is:…

A: We know that a finite group G is said to be cyclic if and only if  there exist an element in G such…

Q: Prove that the group G = [a, b

A: Given, the group G=a, b with the defining set of relations…

Q: 6. Let : (G,*) → (G',') be a group isomorphism, and let a € G. Prove that (a). G is abelian if and…

A:

Q: 6. Let G be a group with the following property: "If a, b, and c belong to G and ab=ca, then b = c."…

A: For a group to be abelian it should follow that ab=ba for all a,b belonging to G. Now we will…

Q: group ⟨a,b,c|b^4 = a,c^(−1) = b⟩ is abelian.

A:

Q: is] Let G and H be groups, and let T:G→H_be Isomorphism. Show that if G is abelian then H is also…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)-1 = a-'b-1 %3D

A: Given: The statement is, let a and b be elements of a group G. Prove that G is abelian if and only…

Q: Prove that a group of order 15 is cyclic

A:

Q: If G is a finite group and some element of G has order equal to the size of G, we ca say that G is:…

A: an abelian group, also called a commutative group, is a group in which the result of applying the…

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

A:

Q: The center of an abelian group G is: a) {e) b) G c) A cyclic subgroup d) None of these

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 (G1, ) and (G2, *) be two groups and o: G1 G2 be an isomorphism. The O G2 might not be abelian…

A: Two group are isomorphic the the properties of groups are must be same, i.e,  if one group is an…

Q: Assume that G is a group such that for all x E G, * x = e. Prove that G is an abelian group.

A: Here we have to prove that G is an abelian group.

Q: If (G, ) is a group with a = a for all a in G then G is %D abelian

A:

Q: If G is a cyclic group of order n, then G is isomorphic to Zn. true or false?

A:

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

A:

Q: Let H be the set of all elements of the abelian group G that have finite order. Prove that H is a…

A: Let H be the set of all elements of the abelian group G that have finite order.  Prove that H is a…

Q: Let G = (1,-1,i,-1} Prove G is a cyclic group under the multiplication operation.

A:

Q: Using the Theorem of Lagrange, prove that a group G of order 9 is abelian

A:

Q: If (G, * ) is a group with a a for all a in G then G is abelian

A:

Q: Prove that a group G is abelian if and only if (ab)-1 = a-lb-l Va,bEG Attach Filo

A:

Q: Let H and K be subgroups of a group G. If |H| = 63 and |K| = 45,prove that H ⋂ K is Abelian.

A: Given: The H and K are subgroups of a group G. If |H| = 63 and |K| = 45 To prove that H ⋂ K is…

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: If G is a group with identity e and a2 = e for all a ∈ G, then prove that G is abelian.

A:

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

A:

Q: prove that a group G of order p^2, where p is a prime, is abelian.

A: Suppose, G is a group of order p2 where p is prime.

Q: '. Assume that G is a group such that for all x E G, x * x = e. Prove that G is an abelian group.

A: Consider any two elements a and b in G. So, a,b,ab,ba∈G. Note that I am directly writing the…

Q: Assume that G is a non-abelian simple group and that |G| < 168. Prove that G = A5.

A: Given: G is a non-abelian simple group and order of G&lt;168.  We have to prove that G≅A5 We will…

Q: Exercise 8.1. Let G₁, G2, ..., Gn be a finite collection of groups. Prove that and only if each G;…

A:

Q: Prove that if G is an abelian group of order n and s is an integer that divides n, then G has a…

A: G is an abelian group of order n ; And, s is an integer that divides n;

Q: If A is an abelian group with A <G and B is any subgroup of G, prove that ANB < AB.

A:

Q: A group G in which (ab)? = a²b² for all a, b in G, is necessarily O Abelian Of order 2 O Finite

A: Since it has associative property (ab) ^2  = a^2b^2

Q: a. Show that (Q\{0}, + ) is an abelian (commutative) group where is defined as a•b= ab b. Find all…

A:

Q: Prove that: Theorem 3: Let G be a group and let a be a non-identity element of G. Then |a| = 2 if…

A:

Q: OLet a and b be elements of a group G. Prove that G is abelian if and only if (ab)- = a¯\b-!.

A: Prove that G is abelian if and only if (ab)-1=a-1b-1. For all a and b be elements of a group G.

Q: Let G be a group. V a, b, c d and x in G, if axb = cxd then ab = cd then G is necessarily:…

A: The answer is given as follows :

Q: Let G be a finite non-abelian simple group and let q be prime, then [G] is

A: It is given that G be any finite non Abelian simple group and q be any prime. We have to determine…

Q: Let G, H, and K be finitely generated abelian groups. Prove or disprove: If G × H ∼= K × G, then H…

A: