Suppose H and K are subgroups of a group G.  If |H|=12 and |K| = 35, find |H intersected with K|.  Generalize.

Q: 4. Let H & K are two subgroups or a group G such that H is normal in G then show that HK is a…

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Q: 1. Give, if possible, one generator for the subgroup H = of Z. Justify your answer.

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Q: 1. Determine the Laplace transform of the function: f(t) = e¬t sinh 2t + e¯2t cos t s - 2 2 | A.…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let G be a group and let a be a fixed element of G. Show that H = {x E G |xa = ax} is a subgroup of…

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Q: let G be a group and H a subgroup of G. prove that for any element gEG holds that gH=H if and only…

A: We can solve the given question as follows:

Q: Prove the next statement: If H is a subgroup of a finite abelian group G, then G has a group…

A: Introduction: In mathematics, an isomorphism is a type of mapping or function between two…

Q: . Suppose that G be a group and H a subgroup of G. Let g be an element of G. Suppose that the left…

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Q: Prove that if H is cyclic, then ø[H] is also cyclic.

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Q: If H is a finite subgroup of a group G and x ∈ G, prove that |N(H)| = |N(xHx-1)|.

A: Given: If H is a finite subgroup of a group G and x∈G. To Prove : NH=NxHx-1

Q: Find all the Sylow 3-subgroups of S4.

A: We have to find all the Sylow 3-subgroups of S4.

Q: 4. Let G be a group of order 14. Using Lagrange's theorem, show that every subgroup of G either…

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Q: Let G be a group and let x ∈ G. Prove that if a has order of 12, then x3 has order 4.

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Q: Find all the subgroups of Z48. Then draw its lattice of subgroups diagram.

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Q: 5.3 Let H be the set of elements () of GL(2, R) such that ad-bc-1. Show that H is a subgroup of…

A: be the set of elements  of  such that  that is the We have to show that  is a subgroup of

Q: Let G be a group and H, K≤ G normal subgroups of G. Prove HnK≤G.

A: Given that  is a group and  are normal subgroups of .We need to prove that  is a normal subgroup.We…

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: Question 2. Suppose that G is a group that has exactly one non-trivial proper subgroup. Prove that G…

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Q: Determine which of the following are groups under the stated operations. For those which are groups,…

A: i) the pair here we draw composition table :246824826486426246886284from above table ,we concluded…

Q: def S ER a ca> Show thats is a subgroup ob (M2(R),+). (6) Show that ib A, B ES then ABES. (C) Find…

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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: Find all the cosets of the subgroup 4Z of Z

A: In the group of integers under addition, the subgroup consists of all multiples of 4, i.e., .A coset…

Q: Please solve and show working. Q: If a nonabelian group G has order 100, show that G has more than…

A: Sol:- We can use Sylow's theorems to approach this problem. Recall that if G is a finite group and p…

Q: 2. Let G be a group. Prove or disprove that Z= {x E G: xg= gx for all g€ G} Isa Subgroup of G.

A: To show Z is a subgroup of G, we need to show that (a) Z is non empty (b) For every a , b∈Z , we…

Q: 4-Let (Z12, +12) be a group and let S={4,6}, find subgroup H generated by S. if exist

A: I have used the definition of subgroup generated by a subset.

Q: In S3, find an example of an H < G, and g1, g2 ∈ G such that the two conjugate subgroups are not…

A: Let H be a subgroup of G Then a H1={ghg | g,h belongs to G,H respectively}. is a conjugate subgroup

Q: Let C be a group and let a be an element of C Prove that = .   Please be clear with theorems ana…

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Q: Take its Suberou P4 graup.of oder sir and order three, then. ৮ वेराण््डयमड jind the distinct left…

A: Take the group G as ℤ6 ( group of integers under addition modulo 6) ℤ6=0,1, 2, 3, 4, 5 Take H=0, 2,…

Q: 19. Let H and K be subgroups of a group G. Prove that gH ngK is a coset of HOK in G.

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Q: Suppose H, and H2 subgroups of the group G. Prove hat H1 N Hzis a sub-group of G. are

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Q: 2.1.12. Corollary Let (H)be a nomal subgroup of the group (G). There is one -one correspondence…

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Q: (7.1) Let (G,) be a group. Prove that the right cancellation law holds in G. Provide reasons in your…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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…

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Q: wwwton. Prove: An intersection of any two subgroups of G is a subgroup of G. ipar satiefies the

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Q: 4. Let G be a group and let H, K be subgroups of G such that |H| = 12 and |K| = 5. Prove that HNK =…

A: We have to prove given result:

Q: 9.18 Suppose Hand K are subgroups of a group G and x, y € G. Prove that if Hx = Ky then H=K.

A: Given:suppose H and K are subgroups of a group G. and to prove:If Hx=Ky then H=K

Q: Let G be a group having two finite subgroups H and K such that gcd(|H.K) 1. Show that HOK={e}.…

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Q: b. Find the center and the commutator subgroup of S2 × Z7.

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