Let G be a abelian group. Prove that the set of all elements of finite order forms a subgroup of G.

Q: abelian group is not cyclic if and only if it contains a subgroup isomorphic to Zp×Zp

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Q: Find a proper subgroup of the group of integers Z under addition and prove that this subgroup is…

A: Consider 2 ℤ = {an / n∈ ℤ }This is a subgroup of ℤ .claim : 2ℤ is isomorphic to ℤ.…

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

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Q: Prove that a group of even order must have an element of order 2.

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Q: Prove that a group of order 12 must have an element of order 2.

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Q: Prove that a group of order 3 must be cyclic.

A: Given the order of the group is 3, we have to prove this is a cyclic group.

Q: Find a proper subgroup of the group of integers ? under addition and prove that this subgroup is…

A: Solution: The objective is to find a proper subgroup of the group of integers and to show that this…

Q: Prove that, there is no simple group of order 200.

A: Solution:-

Q: let G be a group of order p^2 where p is prime. Show that every subgroup of G is either cyclic or…

A: Given that G is a group of order p2, where p is prime.To prove that every subgroup of G is either…

Q: Let G be a finite group. Then G is a p-group if and only if |G| is a power of p. We leouo the

A: Given G is finite group and we have to prove G is a p-Group of and only if |G| is a power of p.

Q: Prove that every subgroup of an abelian group is abelian

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Q: Prove that if G is a finite cyclic group, Hand K are subgroups of G, and H =f. K, then IHI =f. IKI

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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 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: Prove that S, is isomorphic to a subgroup of An+2-

A: An even permutation can be obtained as the composition of an even number and only an even number of…

Q: Let G be a group with |G|=187 then every proper subgroup of G is: * Non cyclic None of these Сyclic…

A: Given that G is a group and G=187 i.e., number of elements in G is 187.  First we have to find the…

Q: Let G be an abelian group. Show that the set containing all elements of G with finite order is a…

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

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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: State the first isomorphism theorem for groups and use it to show that the groups/mz and Zm are…

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Q: = Prove that, there is no simple group of order 200.

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 be a group of finite order n. Prove that an = e for all a in G.

A: Let G be a group of finite order n with identity e.           Since G is of finite order…

Q: Prove that any two Sylow p-subgroups of a group G are conjugate.

A: To prove that any two sylowp-subgroup of a group G are conjugate

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

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Q: Suppose that G is a finite Abelian group that has exactly one subgroup for eah divisor of |G|.  Show…

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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: Let G be a group with center Z(G). Assume that the factor group G/Z(G) is cyclic. Prove that G is…

A: To prove that the group G is abelian if the quotient group G/Z(G) is cyclic, where Z(G) is the…

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: Let G be a group and define the map o : G → G by $(g) = g¬1. Show that o is an automorphism if and…

A: Automorphism and abelian group

Q: Let p : G → G' be a group homomorphism. (a) If H < G, prove that 4(H) is a subgroup of G' (b) If H <…

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Q: Let G be a group. Prove that if G is abelian, then the mapping ø(g)=g¬ for all g e G is an…

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Q: Let G be an Abelian group and let H be the subgroup consisting ofall elements of G that have finite…

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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 Z denote the group of integers under addtion. Is every subgroup of Z cyclic? Why? Describe all…

A: Solution

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

A:

Q: (A) Prove that, every group of prime order is cyclic.

A: Let, G be a group of prime order. That is: |G|=p where p is a prime number.

Q: Let G be a group with |G|=187 then every proper subgroup of G is: * Non abelian Non cyclic None of…

A: See the detailed solution below.

Q: Let (G,*) and (H,*) be finite abelian groups. If G x G = H x H then G=H. Show that they are…

A: Given that, G×G=H×H⇒G=H Since G,* and H,* are both finite abelian groups we get,…

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: Let G be a group such that a^2 = e for each aEG. Then G is * Non-abelian Cylic Finite Abelian

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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 non-trivial group. Prove that Aut(G) × Aut(G) is Aut(G x G). a proper subgroup of

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Q: Prove that every group of order 375 has a subgroup of order 15.

A: According to the given information it is required to prove that every group of order 375 has a…

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…