Suppose that K is a proper subgroup of H and H is proper subgroup of G. If |K| = 42 and |G| = 420, then what are the possible orders of H?

Q: Suppose that H is a subgroup of G and let G act by conjugation on the set X = {gHg-1 : g ∈ G}.…

A: Given that,H is a subgroup of G and G act by conjunction on the set

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

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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…

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

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Q: 10. Let A be a subgroup of G, and let B be a subgroup of H. Show that A×B is a subgroup of G×

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Q: c) Show that if G is a group of order 100, then G has at most one subgroup of order 25.

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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: (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: 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: Suppose that p:G→G'is a group homomorphism. Show that () p(e) = ¢(e') (1) For every gEG, (ø(g))-l =…

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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…

A: To show:

Q: Theorem: Let (K,+)is a subgroup of a group (H, ) and (H,) is a subgroup of a group (G,) then (K, )is…

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Q: If H is a proper subgroup of L and L is a proper subgroup of G. If |G| = 1250 and |L| = 125, list…

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Q: Prove If S1 and S2 are subgroups of G, then S1 intersection  S2 is a subgroup of G.

A: Let S1 and S2 are two subgroups Then if x, y E S1 or S2 .xy E S1 or S2 And V x E S1 or S2 Then x-1 E…

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: 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: Let G be an abelian group and suppose that H and K are subgroups of G. Show that the following set…

A: According to the given information, Let G be an abelian group and suppose that H and K are subgroups…

Q: Let G be a group with |G|=187 then every proper subgroup of G is:

A:

Q: If H and K are subgroups of G, |H|= 18 and |K|=30 then a possible value of |HNK| is

A: It s given that H and K are subgroups of G, H=18 and K=30. Since H, K are subgroups, H∩K≤H  and…

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

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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: Let G = (a) and |a| = 24. List all generators for the subgroup of order 8.

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Q: . Let H and K be normal subgroups of a group G such nat HCK, show that K/H is a normal subgroup of…

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Q: Let H be a subgroup of G and let K=⋂φ∈Aut(G)φ(H). Show that K is characteristic in 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: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

A: Given orders of subgroup 10 18 30

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: Show that the subgroup generated by any two distinct elements of order 2 in S3 is all of S3.

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A: Introduction: If there exists a bijective map θ:G→G' for two given groups G and G', then θ is…

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

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Q: Let H be a Sylow p-subgroup of G. Prove that H is the only Sylowp-subgroup of G contained in N(H).

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Q: 4. Let H be a subgroup of a group G. Show that exactly one left coset of H is a subgroup.

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Q: Q1/ If (H,*) is collection of subgroups of (G,*) then (U H,*) is subgroup of (G,*)

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

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Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right cóset-4 + 2Z contains the…

A: Given the set of all even integers Ififis a subgroup of (Z, +) The right coset is -4 + 2Z

Q: If H and K are subgroups of G, |H|= 16 and |K|=28 then a possible value of |HNK| is

A: It is given that H and K are subgroups of G and H=16, K=28. Since H and K are subgroups of G, H∩K≤H…

Q: 1) If (H, *) is a subgroups of (G, *)then (NG(H) , * ) is a subgroup of (G, *).

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Q: Suppose that G is a group and |G| = pnm, where p is prime and p &gt; m. Prove that a Sylow…

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Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -6 + 2Z contains the…

A: 10 is the element in the right coset.

Q: Let H be a subgroup of G. If g e G, show that gHg-1 = {ghg-1|he H} is also a subgroup of G. This…

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

A: Suppose that, H1 and H2 be the sub groups of G. Also, let H=H1∩H2 . Consider one step group test:…

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Q: Let H be a subgroup of Sn. (a) Show that either all the permutations in H are even, or else half the…

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Q: Let a be an element of a group G such that Ord(a) = 30. If H is a normal subgroup of G, then Ord(aH)…

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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: Suppose that | G|= 20. What're possible orders for a subgroup of G. 35.

A: We can find the subgroup of G as below.