Let G be a group and a E G. The centralizer of a in G is equal to the centralizer of a-l in G. The subgroups of U(8) are all non-cyclic since U(8) is non-cyclic. The set numbers Q and R under addition is a cyclic group. If H and K are subgroups of a group G then Hn K is a subgroup of G. An element a of a group G has order n E z+ if and only if a" = e.

Q: Let G = Z8 x Z6, and consider the subgroups H = {(0, 0), (4, 0), (0, 3), (4,3)} and K = ((2, 2)).…

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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 a group and a E G be a certain fixed element of G. The centralizer of a in G is C(a) = {g €…

A: Hey, since there are multiple questions posted, we will answer the first question. If you want any…

Q: 3. Let G be a group, H a subgroup of G, and g and element of G. (a) Prove that {ghg-1 | h e H} is…

A: Let G be a group, H is a subgroup of G, and g be an element of G

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

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Q: prove that the group G=[a b] with defining set of relations a^3=e, b^7=e, a^-1ba=b^8 , is a cyclic…

A: We need to prove that , group G = a , b  with defining sets of relations a3 = e   ,  b7 = e  also…

Q: Let G be a group. Show that Z(G) = nEC(a). [This means the intersection of all subgroups of the form…

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Q: : Show that if G is a cyclic group then G is abelian. : Let n>0 be a positive integer, and let a and…

A: We will use basic knowledge of group theory to answer this. You have explicitly written on the top…

Q: Let h: G G be a group homomorphism, and gEG is an element of order 35. Then the possible order of…

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Q: Let G be a group and let a e G. In the special case when A = {a}, we write Cg(a) instead of Cg({a})…

A: The given problem is related with group theory. G is a group and a ∈ G. Here, we write Cga instead…

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, and let a E G.  Prove that C(a) = C(a-1).

A: Given: Let G be a group and let a∈G. then we will prove C(a)=C(a-1) If C(a) be the centralizer of a…

Q: Let G be a group and a ∈ G. The centralizer of  a in G is equal to the centralizer of a^-1 in G.…

A: First let us see the definition of centralizer or normalizer of a in G (definition from I.N.…

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: If G is a finite group with |G|<120 and G has subgroups of orders 1O, 15 and 20 then the order of G…

A: Using Lagrange's theorem it can be written that if G is a finite group and B is a subgroup of G,…

Q: Let G be a group. The centre of G is defined as the set Z(G) = {g ∈ G : gx = xg for all x ∈ G}.…

A: Let G be a group. The center ZG of group G is defined as follows. ZG = g∈G : gx = xg , ∀x∈G We need…

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

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Q: An element a of a group G has order n E z+ if and only if a" = e.

A: The given statement is False.

Q: Let G be a group, and let H < G. Assume that the number of elements in H is half of the number of…

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Q: {hk | h ∈ H, k ∈ K}}

A: We have to prove that  {hk|h∈H, k∈K} is a subgroup of G.

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: 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: Prove property 1

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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: . 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: 1. Let (G. *) be a group and a E G. Suppose that a *a = a · Prove or disprove that a must be the…

A: We have to solve given problem:

Q: 8. Let (G,*) be a group, and let H, K be subgroups of G. Define H*K={h*k: he H, ke K}. Show that H*…

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Q: Let G be a group, and a, b € G. Prove that b commutes with a if and only if b- commutes with a.

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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…

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Q: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({a +bv2 : a,…

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Q: Let G be a cyclic group with generator a, and let G' be a group isomorphic to G. If : G -+ G' is an…

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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: Let G be a group and let a e G. The set CG(a) = {x € G | xa = that commute with a is called the…

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Q: If G is a finite group with IG|<160 and G has subgroups of orders 10, 16 and 20 then the order of G…

A: Given :- order of a group G... |G| &lt; 160 Also order of subgroups of group G are 10, 16 , 20. We…

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

A: Using Lagrange's theorem it can be written that if G is a finite group and A is a subgroup of G,…

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…

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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 is the subgroup of group G where G is the additive group of integers and H = {6x | x is the…

A: Let H is a subgroup of order 6 . Take H=6Z where Z is integers.

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: Let G be the additive group R × R and H = {(x, x) : x E R} be a subgroup of G. Give a geometric…

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Q: Let G be a group and a be an element of this group such that a12 = e. The possible orders of a are:…

A: Option (4)

Q: In the group (Z, +), find (-1), the cyclic subgroup generated by -1. Let G be an abelian group, and…

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Q: 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.…

A: Since H is a cyclic group. and K is cyclic group. H=a,K=b If H⊂K then H∩K=H where H is cyclic If K⊂H…

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: 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: G be a group and let a e G have order 30. List the powers of a that have order 2, the rers that have…

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Q: Let G be a group with |G|=221 then every proper subgroup of G is: * Non cyclic Non abelian Сyclic…

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