Let a belong to a group and a12 = e. Express the inverse of each ofthe elements a, a6, a8, and a11 in the form ak for some positive integer k.

Q: If g and h are elements from a group, prove that ΦgΦh = Φgh

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Q: 3. Define an operation on G = R\{0} x R as follows: (a, b) (c,d) = (ac, bc + d) for all (a, b),…

A: 3. Define an operation * on G=ℝ\{0} ×ℝ as follows: (a,b)*(c,d)=(ac, bc+d) for all (a,b), (c,d) ∈G…

Q: Let S9 be the symmetric group and let a be an element of S, defined by: 4 5 6 7 8 9 2 9 867 1 2 3 )…

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Q: Let G be a cyclic group ; G=, then (c*b)^=c4* b4 for all a, c, b EG.

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Q: For any elements a and b from a group and any integer n, prove that (a21ba)n 5 a21bna.

A: For any elements a and b form a group and any integer n, We have to prove that a-1ban=a-1bna.

Q: that a and xax^-1 have same order for all x belongs G.

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Q: If a is an element of order 8 of a group G, and = ,then one of the following is a possible value of…

A: Given that a is an element of order 8 and a4=ak

Q: Let G be the set of nonnegative integers. Is a group, |a - b| for all a, b, EG? where a * b =

A: We will check whether it is group or not.

Q: Which of the following groups are cyclic? For each cyclic group, list all the generators of the…

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Q: Let G be the group with presentation (x,y: 1² = 1, r² = y²₁ xy = y ¹). Decide how many elements are…

A: Consider the given group G with presentation, x,y:x4=1, x2=y2, xy=yx-1 it can be observed that it is…

Q: Show that each of the following is not a group. 1. * defined on Z by a*b = |a+b|

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Q: Let X be a group and we then let x and y be an element of X.  Prove that (x*y)^ -1 = a^-1 * b^-1 iff…

A: Since there are some mistakes in given typed question.question may like "Let X be a group and let…

Q: . Let G be a group and a e G. An element be G is called a conjugate of a if there exists an element…

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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…

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Q: Let G={aeR: a>0}. Select all of the operations from the following list such that (G, ◊) is a group.…

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Q: Let G be a group. Show that for all a,b,c G, (acb-1)-1 = bc-la.

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Q: Let G be a finite group and a€G s.t |a|=12.if H= find all other generators of H.

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Q: Verify that (ℤ, ⨀) is an infinite group, where ℤ is the set of integers and the binary operator ⨀ is…

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Q: Let G be a group and let a be a non-identity element of G. Then |a| = 2 if and only if a = a-1.

A: Let G be a group with respect to * . Let e be the identity element,and a is non identity element.…

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: I denotes the set of real numbers and (*) is an operation on R such that a * B = a +B+ aß for ali a,…

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Q: f H and K are two subgroups of a group G, then show that for any a, b ∈ G, either Ha ∩ Kb = ∅ or Ha…

A: If H and K are two subgroups of a group G, then show that for any a, b ∈ G,either Ha ∩ Kb = ∅ or Ha…

Q: If G is a group and g E G, show that the number of conjugates of g E G is [G : CG(g)]

A: Given G be a group and g∈G be an element. Let Bg be the set of all conjugate elements of g∈G.…

Q: Let a,b be elements of a group G .Assume that a has order 5 and  a^3b=ba^3. prove that ab=ba.

A: Let a,b be elements of a group G .Assume that a has order 5 and a^3b=ba^3. prove that ab=ba.

Q: for every element a of a ((a) Prove that group 6, 2 (G) is a subset of

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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: Let G = {a +b 2 ∈ ℝ │ a, b ∈ ℚ }. Prove that the nonzero elements of G form a group under…

A: Note : In the question, it has to be a + b2 ∈ ℝ instead of a + b 2 ∈ ℝ. So, we are solving the…

Q: Find order of all the elements of the Group = {1, −1, ?, −?} . The binary operator ∗ is defined as ?…

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Q: For any elements, a and b in a group G, a^(5).b = ba and a^(4) = e, then G is an abelian group.

A: Given a and b in a group G. and  a5b = ba  and  a4 = e To prove:- G is abelian group

Q: Given that A and B is a group. Find out if þ: A→B is a homomorphism. If it is a homomorphism, also…

A: Group homomorphism is nothing but a function defined between two groups. The function must be closed…

Q: Given any prime p, we let G be a group of oder 7p⁶. PROVE or DisPROVE that G is not simple

A: Let  G be a group of oder 7p⁶.

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: 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 p and q be distinct prime numbers and set n = pq. Find the number of generators of the cyclic…

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Q: Let a and b be elements in a group G. Prove that ab^(n)a^(−1) = (aba^(−1))^n for n ∈ Z.

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Q: Let (G,*) be a group such that a² = e for all a E G. Show that G is commutative.

A: A detailed solution is given below.

Q: Is the set of numbers described below a group under the given operation? Even integers; addition O…

A: A set is called group on binary operation if a) It is closed b) It is associative c) It has identity…

Q: Find all the generators of the following cyclic groups: (Z/6Z,+), ((Z/5Z)*, ·), (2Z, +), ((Z/11Z)*,…

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Q: If b and c are the inverses of some element a in a group G then a) b =c b) b c b. c) b = kc for some…

A: Let a identity element e.. And b is inverse of a. Then a.b=e And if c is the inverse element of a…

Q: If b and c are the inverses of some element a in a group G then a) b =c b) b #c b. c) b = kc for…

A: We have to state the correct statement about 'b' and 'c' which are the inverse of some element 'a'…

Q: Find the solution set for each of the following with the representation of the group on the number…

A: To find - Find the solution set for each of the following with the representation of the group on…

Q: let G be a group, a,b E G  such that bab^-1 =a^r , for some  r E N, where N are the natural ones,…

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Q: Let K and H be subgroups of a finite group G with KCHCG.lf [G:K] = 12 and [H:K] = 3. Then, [G:H] =…

A: Let , K  and  H  be subgroups of finite group G. Also . K ⊆ H ⊆ G Here , G : K = 12  , H : K = 3 We…

Q: Determine whether the set G is a group under the operation * G={n integer|n is odd}; a*b=a+b

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Q: The inverse of 3 in the group (Z5, o5) is

A: Definition:

Q: Let a,b be elements of S6 (symmetric group) where a=(1,2)(4,5) and b=(1,6,5,3,2).verify that…

A: Given: Let a,b be elements of S6 (Symmetric group) where a=(1,2)(4,5) and b=(1,6,5,3,2).

Q: Let a, b be elements of a group G. Assume that a has order 5 and a³b = ba³. Prove that ab = ba.

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Q: Let a belong to a group and |a| = 100. Find |a98| and |a70|.

A: Given: a=100 we know that am=am

Q: Let G be a group with identity e and a € G. (a) Define |G|, the order of G, and |al, the order of a.…

A: (a) The cardinality or the number of elements in a group is called the order of that group. If group…