et 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. Does this imply that H∩K is cyclic? Prove or give counter example

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 H be a subgroup of a group G and a, bEG. Then be aH if and only if * a-1b e H O None of these…

A: Given H is a subgroup of G. We need to find a necessary and sufficient condition for a belongs to…

Q: ) If G is a group and X is a G-set, then the subset (g in G: for all x

A: To explain the given statement is true or false as,

Q: Let G, (Isisn) be n groups and G is the external direct product of these groups. Let e, be the…

A:

Q: Let H be a subgroup of a group G and a, be G. Then bE aH if and only if * O a-1b eH O ab-1 eH O None…

A: We know that  b∈bH                    (1) We know that  aH = bH if and only if a-1b ∈H…

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

A:

Q: For any subset H of a group G, if ab^-1 is in H for all a,b element of H then H is a subgroup of G.…

A:

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…

A: To show:

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 H be a subgroup of a group G and a, be G. Then be aH if and only if None of these O ab e H O…

A: We know that b∈bH                                    (1) We know that aH = bH if and only if a-1b∈H…

Q: If d divides the order of a cyclic group then this group has a subgroup of order d. Birini seçin: O…

A:

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: A nonempty subset of a group, that is closed under the operation of the group, is a subgroup. Birini…

A: A nonempty subset of a group is a subgroup only if it is a group under the same binary operation.

Q: Let G be a group, and let X be a set. Let I be the intersection of all subgroups of G that contain…

A: Let G be a group and X be a set in G. Suppose I is the intersection of all subgroups of G that…

Q: Let S, be the symmetric group and let a be an element of S, defined by: 1 2 3 4 5 67 8 9 ) B = (7…

A:

Q: It is not possible that, for a group G and H and K are nomal subgroups of G, H is isomorphic to K…

A: Let G be a group and H and K are normal subgroups of G

Q: Let G and H be groups. Let p : G → H be a homomorphism and let E < H be a subgroup. Prove that p(E)…

A: Given: φ:G→H is a group homomorphism and E≤H. To prove: a) φ-1(E)≤G b) If E ⊲ H then φ-1E ⊲ G

Q: Let G be a group and let H and K be subgroups of G. Prove that the intersection of H and K, H n K =…

A: 1. It is given that H∩K=x∈G|x∈H and x∈K Let  x, y∈H∩K ⇒x,y∈H and x,y∈K⇒xy-1∈H and xy-1∈K⇒xy-1∈H∩K…

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: 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: prove that Z(S_3) is the trivial subgroup, that is, the center of the permutation group S_3 is the…

A: prove that Z(S_3) is the trivial subgroup, that is, the center of the permutation group S_3 is the…

Q: Let G be a group and a e G. Show that o(a) = o(a-). order n, then ba also has order n.

A:

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: 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,…

A:

Q: . Let G be a group, let g e G, and let H - G. Suppose that the element Hg E G/H has order n. Show…

A: We have to prove that given statement:

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: Let H and K be two subgroups of a group G. Let HK={ab|a∈H,b∈K}. Then HK is a subgroup of G. true or…

A: F hv

Q: Suppose that G is a finite group and let H and K be subgroups of G. Prove that |HK| = |H||K|/|HN K|.

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Q: Let H and K be subgroups of a group G. Prove that HNK is a subgroup of G.

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Q: Let H and K be subgroups of the group G, and let a, b E G. Show that either aH n bK = Ø or else aH N…

A:

Q: Let G be a group and let A1, A2, B1, B2 be the proper nontrivial subgroups of G. Suppose that A1×A2…

A:

Q: Let H be a subgroup of a group G and a, be G. Then b E aH if and only if ab-1 e H O ab e H O None of…

A: Ans is given below

Q: Suppose the o and y are isomorphisms of some group G to the same group. Prove that H = {g E G| $(g)…

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Q: A subset of H of a group G is a subgroup of G if the operation on G makes H into a group. Prove that…

A:

Q: Let G be a group and g E G. Prove that if H is a Sylow p-group of G, then so is gHg-1

A: It is given that, G is a group and g∈G. To sow that if H is a sylow p-subgroup of G, then so is…

Q: a) Show that given a finite group G and g ∈ G, the subgroup generated by g is itself a group. (b)…

A:

Q: If G is a finite group with |G|<180 andG has subgroups of orders 10, 18 and 30 then the order of G…

A: Use the fact that order of subgroup divides order of group

Q: If H and K are subgroups of a group G then H n K is a subgroup of G.

A: Note:  according to our guidelines we can answer first question and rest can be reposted.   Lemma:…

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

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Q: If a subgroup H of a group G is cyclic, then G must be cyclic. Select one: O True O False

A: we will give the counter example in support of our answer.

Q: 2) Let (G, *) be a group and H, K be subgroups in G. Prove that subset H * K is a subgroup if and…

A:

Q: Although (H,*) and (K,*) are subgroup of a group (G,) then (H * K,*) may field to be subgroup of (G,…

A:

Q: Let H and K be subgroups of a finite group G. Show that |HK |HK= |HОКI where HK (hk hE H, k E K}.…

A: let D = H ∩K then D is a subgroup of k and there exist a decomposition of k into disjoint right…

Q: Prove that ifH and K are subgroups of a group G with operation *, Question 8. then HNK is a subgroup…

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Q: i have included a picture of the question i need help understanding.thank you in advance. please…

A: Let H and K are two subgroups of the group G.To show

Q: Suppose that G is a group such that Ord(G) = 36. The number of subgroups that G has is 4 O 12 O 18…

A: Given order of G is 36 So U(G) = {1,5,7,11,13,17,19,23,25,29,31,35} So number of elements are 12…

Q: Suppose that G is a group such that Ord(G) = 36. The number of subgroups %3D that G has is 4 О 12 О…

A: Order of a group: Let G be a group and n be the number of elements in the group. Then, order of…

Q: Let c and of d be elements of group G such that the order of c is 5 and the order of d is 3…

A: Need to find intersection of subgroup