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 3)(1 5 4). a = 5 4 3 2 9 8 67 1 Find the smallest integer n such that an+3 = a.

Q: 1. Let a and b be elements of a group G. Prove that if a E, then C. 2. Let a and b be elements of a…

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Q: 1. Let G be a group and H a nonempty subset of G. Then H <G if ab-EH whenever a,bEH

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Q: Let G be a group of order pm where p is a prime number and m is a positive integer. Show that G…

A: Let G be a group of order pm, where p is a prime number and m is a positive integer. Then,…

Q: Let x be an element of group G. Prove that if abs(x) = n for some positive integer n, then x-1 =…

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Q: 4. If a is an element of order m in a group G and ak = e, prove that m divides k. %3D

A: Step:-1 Given that a is an element of order m in a group G and ak=e. As given o(a)=m then m is the…

Q: 2. Let G be a group and H1, H2 <G subgroups. (a) Suppose |H1| = 12 and |H2 = 28, prowe H1 n H2 is…

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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 G be a group and let a, b E G. (a) Prove that o(ab) = o(ba). (Note that we are not assuming that…

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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: (G,+) is a finite group such that (a+b)^2 = a^2 + b^2 for all a,b E G. show that G is abelian

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Q: Let f be the following permutation: 1 2 3 4 5 6 7 f = 4 3 6 2 1 5 7 a. Determine the group to which…

A: Given permutation is f=12345674362157 f is permutation of 7 elements so it is belong to s7. i.e. sn…

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: Show that if G is a finite group of even order, then there is an a EG such that a is not the…

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Q: If G is a finite group with |G|<160 and G has subgroups of orders 1O, 16 and 20 then the order of G…

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Q: Exercise 3.3.2 Find all left cosets of the subgroup H={1, 11 (mod 20)} in the multiplicative group…

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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 G be a finite group. Then G is a p-group if and only if |G| is a power of p. We leouo the

A: Given G is finite group and we have to prove G is a p-Group of and only if |G| is a power of p.

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 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: c) Show that Z,,+, is a cyclic group generated by 3

A: 3(c) To check if 3 is generator of (Z5 , +5) , we must check that 3 generates all the members of Z5…

Q: Compute all generators 1) of the multiplicative group Z'n 2) of the multiplicative group Z's.

A: (1)

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: 3.) Prove that for every element a of a Group G, Z(G) is a subset of C(a)

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Q: Let l be an integer greater than 1, and let G be a finite group with no element of order l. Can…

A: Find the attachment for the solution.

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: If A is a group and B is a subgroup of A. Prove that the right cosets of B partitions A

A: Given : A be any group and B be any subgroup of A. To prove : The right cosets of B partitions A.

Q: Q3: (A) Prove that 1. There is no simple group of order 200.

A: Simple group of order 200

Q: Suppose that a is an element of a group G. Prove that if there is some integer n, n notequalto 0,…

A: Suppose that a is an element of a group G. Prove that if there is some integer n, such that n≠0 and…

Q: Q\ Let (G,+) be a group such that G={(a,b): a,b ER}. Is ({(0,a): aER} ,+) sub group of (G,+).

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Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 ano then the order of G is:…

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Q: (b) Given two groups (G,) and (H, +). Suppose that is a homomorphism of G onto H. For BH and A:{g…

A: Given that G,·and H,* are two groups.

Q: Let (G1, +) and (G2, +) be two subgroups of (R, +) so that Z+ ⊆ G1 ∩ G2. If φ : G1 → G2 is a group…

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Q: Let G be a group and a be an element of this group such that a^63e. The possible orders of a are: *…

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Q: Let G be a finite group. Let xeG, and let i>0. Then prove that o(x) gcd(i,0(x))

A: To prove the required identity on the order of the group element

Q: Show that if p and q are distinct primes, then the group ℤp × ℤq is isomorphic to the cyclic group…

A: We have to show that if p and q are distinct primes, then the group Zp×Zq is isomorphic to the…

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

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Q: Show that the quotient group Q/Z is isomorphic to the direct sum of prufer group

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Q: 2. For an arbitrary set A, the power set P(A) = {X | X C A}, and addition in P(A) defined by X+ Y =…

A: Recall the following. Definition of group: Suppose the binary operation * is defined for element of…

Q: Let G be a group and a e G such that o(a) = n < oo. Show that a = a' if and only if k =l mod n. %3D

A: Let G be a group and a∈G such that Oa=n&lt;∞. Show that ak=al if and only if k≡l mod n. If k=l the…

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 G be a group. V a,b,c d and x in G, if axb=cxd then ab=cd then G is necessarily: O Abelian O Of…

A: Solution:Given    G be a group∀a,b,c,d and x in G

Q: Let G be a group, and assume that a and b are two elements of order 2 in G. If ab = ba, then what…

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Q: 64

A: Under the given conditions, to show that the cyclic groups generated by a and b have only common…

Q: Let G be a group, a E G. Prove that a=a + a < 2

A: Concept:

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 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…

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

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Types of Bias

Statistics is the study of data collection, organization, analysis, interpretation, and presentation. Statistical bias is a characteristic of a statistical technique or its findings in which the expected value deviates from the actual root quantitative parameter being estimated. According to the actual definition of bias, it refers to the tendency of a statistic to overestimate or underestimate a parameter.