A group G has order 4n. where n is odd. Show that G has no subgroup of order 8.

Q: If p is a prime, prove that any group G of order 2p has a normal subgroup of order p and a normal…

A: To prove that any group of order 2p has a normal subgroup of order p and a normal subgroup in g

Q: Consider the dihedral group D6.Find the subgroups of order 2,3,4 and 6

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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: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -3 + 2Z contains the…

A: Given that 2Z is the set of all even integers.

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: C. Find all subgroups of the group Z12, and draw the subgroup diagram for the subgroups.

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Q: A simple group is called G if G has no ordinary subgroup other than itself, and suppose f: G → H is…

A: The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. If we…

Q: 2) Let be H. K be and gooup Subgroups f Relate Gu such That Na(H)=Nq(K). H and 'K.

A: Let G be a group. Let H and K be a subgroups of G such that NG(H)=NG(K) We relate H and K. Let G be…

Q: Given that G is a group and H is a subgroup. What is the result of (b^-1)^-1 if b is an element of…

A: Given that     G is a group and H is a subgroup of G.    Inverse of an element:  Let G be a group…

Q: a The group is isomorphic to what familiar group? What if Z is replaced by R?

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Q: There is a group G and subgroups A and B of orders 4 and 6 respectively such that A N B has two…

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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: If H and K are two subgroups of finite indices in G, then show that H ∩ K is also of finite index in…

A: If H and K are two subgroups of finite indices in G, then show that H ∩ K isalso of finite index in…

Q: Every finite group of order 36 has at most 9 subgroups of order 4 and at most 4 subgroups of order 9…

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Q: Let G = A4, the alternating group of degree 4. (a) How many elements of order 3 does G have? (b)…

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Q: If G is a finite group with |Gl<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

A: Given: The group G is a finite group with | G | &lt; 120 and G has subgroups of orders 10, 15, and…

Q: Consider find Subgraup. Dihedral group D- of order 2,3,4 and 6.

A: A group G of two generators x and y of order n and 2 respectively with some relation is called the…

Q: If N is a normal subgroup of order 2 of a group G then show that N CZ(G).

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Q: Let G = Z, be the cyclic group of order n, and let S c Z, \ {0}, such that S = –S, |S| = 3 and (S) =…

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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: Theorem(7.11) : If (H, *) is a subgroup of the group (G, *) , then the pair (NG(H), *) is also a…

A: The normalizer of G, is defined as,  NG(H) = { g in G : g-1Hg = H }

Q: let H be a normal subgroup of G and let a belong to G . if the element aH has order 3 in the group…

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Q: The subgroup {e} is called the nontrivial, that is, a subgroup that is not e is nontrivial.…

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Q: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({3* : k E…

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

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

A: Suppose G is a group and H is a subgroup of G .Then set a+H is left coset if H in G . set H+a is…

Q: et G be a group and suppose that x E G has order n. Let d be a divisor of n. Show that G as an…

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Q: Show that every group of order (35)3 has a normal subgroup of order 125

A: Given, A Sylow 5-subgroup of a group of order 353 is of order 125. The divisors of 353 that are not…

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: 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: Suppose that G is a finite simple group and contains subgroups Hand K such that |G:H| and |G:K| are…

A: Consider the finite simple group G that has subgroup H and K. |G: H| and |G: K| are relatively…

Q: Let G = Z, be the cyclic group of order n, and let S c Z, \ {0}, such that S = -S, \S| = 3 and (S) =…

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Q: Suppose that a subgroup H of S5 contains a 5-cycle and a 2-cycle.Show that H = S5.

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Q: Let G Są and let K = {1,(1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}. K is a normal subgroup of G. What is…

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

A: 7 is the correct answer.

Q: If G is a group with 8 elements in it, and H is a subgroup of G with 2 elements, then the index…

A: We are provided that a group G with 8 elements and H is a subgroup of G with 2 elements and…

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 having two finite subgroups H and K such that gcd(|H.K) 1. Show that HOK={e}.…

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Q: For A, the alternating subgroup of S, show that it is a normal subgroup, write out the cosests, then…

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

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Q: Let G be a group and let a E Ga G with a = 8. the order of a² is not equal to the order of the…

A: The given statement is

Q: 5. If H. aEA are a family of subgroups of the group G, show that is a subgroup of G.

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Q: Suppose H and K are subgroups of a group G. If |H| = 12 and |K| = 35, find |H N K|. Generalize. %3D

A: Given that H and K are subgroups of a group G. Also, the order of H is H=12 and the order of K is…

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: 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…

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.