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

Q: 3. Prove that a subset H of a finite group G is a subgroup of G if and only if a. His nonempty, and…

A: We have to prove given property:

Q: Prove that, if H is a subgroup of a cyclic group G, then the quotient group G/H is also cyclic.

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Q: Show that if H and K are subgroups of an abelian group G, then {hk|h € H and k e K} is a subgroup of…

A: A set G is called a group if it satisfies four properties  Closure property:  ab∈G  where a,b ∈G…

Q: Let G be an Abelian group and H = {x E G | |x| is odd}. Prove thatH is a subgroup of G.

A: Given: To prove H is a subgroup of G.

Q: Prove that GL(2,R) is not an abelian group

A: Solution is given below:

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

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

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

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

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Q: Prove that E(n) = {(A, ¤) : A e O(n) and E R"} is a group. %3D

A: Consider the given: E(n)={(A,x)} where A∈O(n)and x∈ℝn

Q: Let G be a group. Show that for all a, b E G, (ab)2 = a2b2 G is abelian

A: To prove that the group G is commutative (abelian) under the given conditions

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

A:

Q: Show that if H is any group and h is an element of H, with h" = 1, then there is a unique…

A: Given that H is a group and h ∈H  Now,we define a mapping f:Z→H  such that   f(n) = hn for n∈Z For…

Q: Suppose G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G,…

A: Let G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G

Q: Suppose that the fundamental group of X is Z and p(xo) is finite. Find the fundamental group of X.

A: Given fundamental group of X is  ℤ and p-1(xo) = finite value Now we have to find the fundamental…

Q: Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group {e, a, b, a…

A: Given :              A group of order 4  Corollary :                    The order of an element of a…

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: 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: 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: Let G be any group with the identity element e. With using the Group Homomorphism Fundamental…

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Q: If A is a group and B is a subgroup of A. Prove that the right cosets of B partitions A

A: Given : A be any group and B be any subgroup of A. To prove : The right cosets of B partitions A.

Q: 2. Let (G. .) be a group such that a.a = e for all a EG. Show that G is an abelian group.

A: Definition of abelian group : Suppose <G, .> is a group then G is an abelian if and only if…

Q: If f: G to H is a surjective homomorphism of groups and G is abelian, prove that H is abelian.

A: As we know that a group homomorphism f:G to H is a map from G to H satisfying:

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 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: Prove that each of the following sets, with the indicated operation, is an abelian group. (a) (R, *)…

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Q: Let (G, -) be an abelian group with identity element e Let H = {a E G| a · a · a·a = e} Prove that H…

A: To show H is subgroup of G, we have show identity, closure and inverse property for H.

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

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

A: A group G is abelian if it is commutative under the operation *. In other words, G,* is an abelian…

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

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Q: Let G be a group, and N ⊆ Z(G) be a subgroup of the center of G, Z(G). If G/N, the quotient group is…

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

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Q: Let x be in a group G. If x' - e and x* - e , prove that x - e and x' = e

A: Let G be a group and x∈G.Given: x2≠e and x6=e , where e is the identity element.To Prove: x4≠e and…

Q: Let H be a subgroup of a group G with a, b ϵ G. Prove that aH= bH if and only if a ϵ bH.

A: For the converse, assume a-1b∈H, we want to show aH=bH Let a-1b=h for h∈H. Suppose x∈aH. Let x=ah1…

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

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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: If G is a non-abelian group of order 8 with Z(G) {e}, prove that |Z(G)| = 2.

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Q: If H is a subgroup of a group G such that (aH)(Hb) for any a, b eG is either a left or a right coset…

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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…

Q: Let G be an abelian group, then (acba)(abc)¯1 is

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Q: Let x belong to a group. If x2e while x : x + e and x + e. What can we say about the order of x? =…

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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: 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: Let G be a group, a E G. Prove that a=a + a < 2

A: Concept:

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

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Q: Let G be a group and suppose that (ab)2 = a²b² for all a and b in G. Prove that G is an abelian…

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Q: On Z is defined the following binary operation; x . y = x + y - 1 Show that (Z, . ) is an abelian…

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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:…

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