Prove that if G is a finite group and H is a proper normal subgroupof largest order, then G/H is simple.

Q: ) Let G be a finite group , IGI=ps. p prime Prove that G cannot have two distinct and sep. subgroups…

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Q: Suppose that a group G of order 231 has a normal subgroup N of order 11. Then, G/N is cyclic O False…

A: Given that G is a group of order 231 and N is an normal sub-group of G of order 11. To show: G/N is…

Q: . Let N be a finite group and let H be a subgroup of N. If [H| is odd and [N:H] = 2, prove that the…

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Q: Prove that every group of order 1225 has a normal abelian Sylow 5-subgroup.

A: Since not a particular question asked as per guidelines solution to only first question  is given…

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

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Q: Q2: Let (G,) be a commutative group, and let the set H consist of all elements of G with finite…

A: Given a group G and a set H of G with the given conditions. We need to show that H is a normal…

Q: Let N be a normal subgroup of a finite group G. Use the theorems ofthis chapter to prove that the…

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Q: Prove that if N is a normal subgroup of G, and H is any subgroup of G, then H ∩ N is a normal…

A: To Prove If N is a normal subgroup of G, and H is any subgroup of G, then H ∩ N is a normal subgroup…

Q: Let G be a finite group, let H be a subgroup of G and let N be a normal subgroup of G. Prove that if…

A: Given that, Let G be a finite group, let H be a subgroup of G and let N be a normal subgroup of G.

Q: If K is a normal subgroup of a finite group G and S is a Sylow p-suby

A: Given that if K is a normal subgroup of a finite group  G and S is a Sylow p-subgroup of G. then K∩S…

Q: If H is a cyclic subgroup of a group G then G is necessarily cyclic * True False

A: let H be a cyclic subgroup of a group G.

Q: 6. If G is a group and H is a subgroup of index 2 in G; then prove that H is a normal subgroup of G:

A: I have proved the definition of normal subgroup

Q: Suppose that G is a group and |G| = pnm, where p is prime andp >m. Prove that a Sylow p-subgroup…

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Q: Prove that the intersection of two subgroups of a group G is a subgroup of G.

A: We will prove the statement.

Q: 9. Prove that if G is a group of order 60 with no non-trivial normal subgroups, then G has no…

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Q: Let N be a finite group and let H be a subgroup of N. If |H| is odd and [N:H] = 2, prove that the…

A: According to the given information, Let N be a finite group and H be a subgroup of N. H is odd and…

Q: Let G be a group of order 100 that has a subgroup H of order 25.Prove that every element of G of…

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Q: Suppose H is a distant and normal subgroup of a group G. Prove that each subgroup of H is a normal…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: 12. Prove that the intersection of any family of normal subgroups of a group (G, *) is again normal…

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Q: Let G be a finite group and H a subgroup of G of order n. If H is the only subgroup of G of order n,…

A: Given, G is a finite group and H is a subgroup of G of order n.

Q: Suppose that G is a finite Abelian group that has exactly one subgroup for each divisor of the order…

A: Here given that G is a finite Abelian group that has exactly one subgroup for each divisor of the…

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: Suppose S is a nonempty subset of a group G.(a) Prove that if S is finite and closed under the…

A: (a)Suppose S is a non-empty subset of a group G. then we have to prove that if S is finite and…

Q: If a cyclic group T of G is normal in G; then show t subgroup of T is a normal subgroup in G

A: Given: A cyclic group T of G is normal in G.

Q: Let (G,*) be an a belian group, if (H,*) and (K,*) are subgroup of (G,*) then (H * K,*) is a…

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Q: Suppose that G is a group with more than one element and G has noproper, nontrivial subgroups. Prove…

A: G is a group with more than one element and G has noproper, nontrivial subgroups. We need to prove…

Q: 5. Suppose G is a group of order 8. Prove that G must have a subgroup of order 2.

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Q: If N is a normal subgroup of a group G, and if every member of N and G/N have a finite order, prove…

A: Given: If N is a normal subgroup of a group G, and if every member of N and GN have a finite order…

Q: Prove that a simple group cannot have a subgroup of index 4.

A: We will prove this by method of contradiction. Let's assume that there exists a simple group G that…

Q: Give an example of a finite group G with two normal subgroups H and K such that G/H = G/K but H 7 K.

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Q: 9. Let (G,*) be a finite group of order pq, where p and q are prime numbers. Prove that any non…

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Q: If H is a cyclic subgroup of a group G then G is necessarily cyclic * O True False

A: this is false because this is need not be true because Z4×Z6 Is not cyclic but have

Q: Suppose that G is a finite group and that Z10 is a homomorphicimage of G. What can we say about |G|?…

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Q: Prove that if H is a normal subgroup of G of prime index p. (Note G can be finite or infinite…

A: It is given that, H is a normal subgroup of G of prime index p. (Here G can be a finite or infinite…

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: If a simple group G has a subgroup K that is a normal subgroup oftwo distinct maximal subgroups,…

A: Here given G is simple group and K is a normal subgroup of G. Then use the definition of simple…

Q: Exercise 7.16. Prove that if N is a normal subgroup of the finite group G and ged(|N|, |G/N|) = 1,…

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Q: Let let G N Subgroup be be of G a a group and normal of finite

A: To prove that H is contained in N, we first prove this: Lemma: Let G be a group.H⊂G. Suppose, x be…

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: Let G be a finite group of order 4 containing no element of order 4. Explain why every nonidentity…

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Q: Show that if G is a group of order 168 that has a normal subgroup oforder 4, then G has a normal…

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Q: Prove that if G is a group of order 60 with no non-trivial normal subgroups, then G has no subgroup…

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Q: If N is a normal subgroup of G and G/N is abelian. Then G is also abelian. Select one: True False

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

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Q: Let G be a group with order n, with n> 2. Prove that G has an element of prime order.

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Q: Show that if H and K are subgroups of an abelian group G then {hk: h element of H and k element of K…

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