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

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

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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:…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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: (G,+) is a finite group such that (a+b)^2 = a^2 + b^2 for all a,b E G. show that G is abelian

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Q: let G be an abelian Group and let H and K be a susb group of G prove that HK ={hk / h an element H…

A: Consider the provided question, Your question is missing the last statement; I think you want to…

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

A:

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: If G is abelian group and (m,n e G) then n'mn:

A:

Q: 12. Prove that the following groups are not cyclic: (a) Z2 x Z2 (b) Z2 x Z (c) Z x Z.

A: To show the following groups are not cyclic. If a group is cyclic the there exit an element in that…

Q: b. Find all abelian groups, up to isomorphism, of order 360.

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Q: Let G be any group and p: G -G, (g) =g1 Vg eG. Then: (a) o is a homomorphism (b) o is a homomorphism…

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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: 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: Let G be a finite abelian group

A: Let d = gcd(m; n).We shall use induction on d.Assume  first that d = 1.

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: * O Abelian O…

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Q: Let G be a group and let a be a non-identity element of G. Then |a| = 2 if and only if a = a-1.

A: Let G be a group with respect to * . Let e be the identity element,and a is non identity element.…

Q: Suppose G is a group of order 48, g € G, and g" = €. Prove that g = ɛ.

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Q: Prove that a group G is abelian if and only if (ab)-1 = a-lb¬ va,bEG

A: We need to prove that a group G is abelian if and only if (ab)-1=a-1b-1 , for all a,b in G. The…

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

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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: Prove that a group G is abelian if and only if (ab)-1 = a¬b¬1 va,bEG

A:

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

A:

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: 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: 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: Prove that (ab)2 = a²b² for all a, b in a group G if and only if G is Abelian.

A: Let G be set and "·" be a binary operation then G,· is a group if  Clouser property. That is, if…

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

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

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

A:

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: 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: Construct the Cayley table for (Zo) ,c), and verify that this is an Abelian group.

A: 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 0 2 3 4 5 6 7 8 0 1 3 4 5 6 7 8 0 1 2 4 5 6 7 8 0 1 2 3…

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

A: Solution:Given    G be a group∀a,b,c,d and x in G

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.

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Q: Prove that a group G is Abelian if and only if (ab)-1 = a-1b-1 forall a and b in G.

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

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…