Let G be a finite group and let primes p and q ≠ p divide |G|. Prove that if G has precisely one proper Sylow p-subgroup, it is a normal subgroup, so G is not simpl

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: Let G be a group and let H and K be subgroups of G so that H is not contained in K and K is not…

A: Given that G be a group and H and K are two subgroup s.t H is not contained in K and  K is not…

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

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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…

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Q: Let G be a nonabelian group. If H and K are cyclic subgroups of G, does it follow that H ∩ k is also…

A: Given that G is non abelian group. Given that H and K are cyclic subgroups of G. The cyclic group…

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: 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: set G be a finite group, P E Syl,(G), and N = (a) Show that Pe Syl, (N). Jumber and Conjugacy of…

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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: Suppose that G is a group of order 168. If G has more than oneSylow 7-subgroup, exactly how many…

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Q: Assume that (G, ) is a group and that (H, ) and (K, ) are subgroups of (G,*). Prove that (HnK,*) is…

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Q: 5. Let p and q be two prime numbers, and let G be a group of order pq. Show that every proper…

A: We have to prove that: Every proper subgroup of G is cyclic. Where order of G is pq and p , q are…

Q: (a) Prove that if K is a subgroup of G and L is a subgroup of H, then K x L is a subgroup of G x H.

A: The detailed solution of (a) is as follows below:

Q: Show that if H and K are subgroups of a group G, then their intersection H ∩ K is also a subgroup of…

A: Subgroup Test           A subset H C G of the group G will be a subgroup if it satisfies the…

Q: Let G be a group, let H be a proper subgroup of G, and let a E G. Prove that the left coset aH is a…

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Q: 32. If H and K are subgroups of G, show that Hn K is a subgroup of G. (Can you see that the same…

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Q: If H is a Sylow p-subgroup of a group, prove that N(N(H)) = N(H).

A: Let G be a finite group and H be the subset of G. Then, normalizer of H in G, when we conjugate H…

Q: Let G be a cyclic group and H be a subgroup of G. Prove that H is a normal subgroup in G and G/H is…

A: Given:- H be a normal subgroup of a group  G⇒gHg-1=H∀g∈G Also given GH  is a cyclic and we need to…

Q: PROVE THAT A SUBGROUP H OF A GROUP G IS NORMAL IF AND ONLY IF gHg^-1=H FOR ALL g is an element G

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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, H,K ≤ G such that H=, K=for some a,b∈G. That is H and K are cyclic subgroups of G.…

A: Given that G is a group and H, K are subgroups of G with the condition that H=<a> and…

Q: Let G be a finite group, p prime, and Ha Sylow p-group. Prove that H is normal in G if and only if H…

A: Given, G is a finite group and H is a sylow p-group o(G)=n, where n is a finite number. p/o(G) from…

Q: If H is a normal subgroup of a finite group G and |H| = pk for someprime p, show that H is contained…

A: H is a normal subgroup of a finite group G and |H| = pk for some prime p.

Q: Suppose that H is a subgroup of a group G and |H| = 10. If abelongs to G and a6 belongs to H, what…

A: Given: H is a subgroup of a group G and |H| = 10 To find: If a belongs to G and a6 belongs to H,…

Q: . Let Sn be the symmetric group on n elements, let An Sn be the alternating subgroup, consisting of…

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Q: Let G be a finite p-group of order p". Show that for all 0<kSn, there is a subgroup order p and each…

A: Given: Let G be a finite p-group of order pn. We have to prove for all 0≤k≤n there is a subgroup of…

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: Let G be a group and suppose that G only has finitely many distinct subgroups. Show that |G| < ∞.

A: If possible  let G is of infinite  order .In particular let  G = {a1,a2,...,an,...} .Let us consider…

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: Let K be a Sylow p-subgroup of a finite group G. Prove that if x ∈ N(K) and the order of x is a…

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Q: a. Prove or Disprove. If H and K be normal subgroups of a group G and H is isomorphic to K, then G/H…

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

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Q: Let G be a finite group. Let E G and let xG be the conjugacy class of x. Prove that x| < |[G, G]],…

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Q: Show that if aH=H then a belongs to H. H is a subgroup of a group G and a is an element of G

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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 G is a group and g E G, the centralizer of g E G, is the set CG(g) := {a E G : ag =ga} that is,…

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Q: A proper subgroup H in a group G is called a maximal subgroup if there is no proper subgroup K of G…

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Q: 189. Let be given Ga finite group and Pe Syl,(G). Give an example of a subgroup H of G where HnP is…

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Q: Let G be a group of order pq where p and q are distinct primes andp &lt; q. Prove that the Sylow…

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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 every group of order 675 has a normal Sylow 5-subgroup.

A: Theorem: If number of Sylow p-subgroup of a group G is unique, then that subgroup is normal. Let G…

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

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Q: G is a finite group of order IGI =pqr with p< q <r prime

A: Solution

Q: If G is a cyclic group, prove for subgroup N that G is a cyclic N

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Q: Let G be a finite group and let H be a normal subgroup of G. Provethat the order of the element gH…

A: Given: G be a finite group and H be a normal subgroup of G.

Q: Prove that the order of the group element gN in G/N divides the order of g.

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Q: Let G be a group with no proper, nontrivial subgroups and assume that |G| > 1. Prove that G must be…

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Q: Let G be a group of order 24. Suppose that G has precisely one subgroup of order 3, and one subgroup…

A: Theorem : If a group G is the internal direct product of subgroups H and K, then G is isomorphic to…