The following is a Cayley table for a group G. The order of 4 * 5 is: 1 3 5 2 3 4 5 2 3 4 2 3 4 5 1 3 4 5 1 2 3 4 5 1 2 3 4 5 4 2 21 5.

Q: In the following Cayley table for a group G, C(2)= 1 2 3 5 2 3 5 1 2 3 4 4 3 4 5 4 5 1 3 4 1 2 3 5 O…

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Q: The following is a Cayley table for a group G. 2.5.4 = 2| 1 2 3 4 1 3 4 1 3 4 1 2 3 3 4 1 4 1 2 1 3…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: The following is a Cayley table for a group G. The order of 3 is: 1 2 3. 4 2 3. 4 5 1 2 3 4 5 2 3 4…

A: In given Cayley Table :            * 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1…

Q: H and K are subgroup of a group G, then which of the following is a group? a) HUK b) HK c) HnK d)KH.

A: Subgroup: Let (G,*) be a group and H be a non-empty subset of G; then, H is called subgroup if H…

Q: Determine whether the following is a semigroup, monoid, or a group. 1. G=Z, a*b = a + ab 2. G=2Z,…

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Q: %24 W NI 52 The following is a Cayley table for a group G2 3 4 12 12 3 2 1 31 4 4 3 O 2

A: To find the 2*3*4

Q: The following is a Cayley table for a group G. The order of 4 is: 1 2 3 4 1 3 4 5 4 4 5 2 4 1 2 3 4…

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Q: Which of the following is the identity element in the group Z/14Z × µs O (147,0) O (142,1) O…

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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: Are groups Z×10and Z×12 isomorphic

A: Concept:

Q: In the following Cayley table for a group G, C(2)= 1 2 2 3 4 5 3 4 5 2 3 4 5 2 3 4 1 2 3 1 2 3 4 5 2…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: In the following Cayley table for a group G, C(3}= 2 3 4 3 4 5 4 1 3 4 1 2 3 2 3 4 4 2 4 O {1,2,4} G…

A: If G be a group and a belongs to G then c(a) = { x : ax = xa , where x belongs in G}.

Q: We will now consider three irreducible representations for the permutation group P(3): E A B I : (1)…

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Q: In the following Cayley table for a group G, C(2)= 2 2 TIT

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Q: If G is a group and a1, a2,…, an shows that a1 * a2 *… * an is unique, regardless of the order in…

A: We are given that G is a group. (G satisfies all the group axioms). Suppose * is the defined binary…

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: The following is a Cayley table for a group G. The order of4 is: 2 1 3 3 4 1 4 1 4 1 5. 2 512 m 45…

A: By observing the Cayley table :

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: The following is a Cayley table for a group G. The order of 3 is: 2 4 2 4 2 3 4 1 3 4 5 2 4 ere to…

A: Given Cayley table

Q: Is S3 x S3 group (the direct product of symmetric group S3) nilpotent?

A: Given question:   Is S3 x S3 group (the direct product of symmetric group S3) nilpotent?

Q: 10) Which of the following is a group? * O (Z,*), a* b = a + b - 1 va, b e Z O (Z,*), a* b = a - 2b…

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Q: Which of the following is a group? O O

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Q: If H and K are subgroups of the group G, then which one of the following is also a subgroup of G? O…

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Q: Let E = Q(√2, √5). What is the order of the group Gal(E/Q)?What is the order of Gal(Q(√10)/Q)?

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Q: Explain the following statement "If G is a group an a E G then o(a) = | |." 31. %3D

A: Order of element a ∈ G is the smallest positive integer n, such that an= e, where e denotes the…

Q: The following is a Cayley table for a group G. 2*5 4 = 1 1 3 1 4 2 4 4. 4. 5. 4. ENG pe here to…

A: 1

Q: order 8 of a group G, and =

A: Given that order of a is 8 .Then  a8=e Rearrange  a little bit , we can have a42=e Hence order of…

Q: In the following Cayley table for a group G, C(3)= 2| 1 3. 4 1| 2 3 4 5 2 3 4 1| 2 3 4 5 1 2 4 5 1 2…

A: The G = {1, 2, 3, 4, 5} 3 * 1 = 4 = 1 *3 3 * 2 = 5 = 2 * 3 3 * 3 = 1 = 3 * 3 3 * 4 = 2 = 4 * 3 3 * 5…

Q: Let G be a group. Let x,y e G be such that O(x) = 4, O(y) = 2, x^3 y = yx. Then O(xy) is 6. Infinity…

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Q: Which of the following is the only trivial sub- group of a group G = {e, a,b, c}? {e,b} {e, a, b, c}…

A: We have to check

Q: 10) Which of the following is not a group? * (Z,*), a* b = a + 2b Va, b E Z O (Z,*), a* b = a +b Va,…

A: Option (1) is correct.

Q: Suppose that <a>, <b> and <c> are cyclic groups of orders 6, 8, and 20,…

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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: The set A := {(1), (1 2 3), (2 3 4)} forms a subgroup of the permutation group (G,.). O True O False

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Q: 1. There is no simple group of order 200.

A: Solution:-As per guidelines I submit first question only

Q: Which of the following is the only trivial sub-group of a group G = {e, a, b, c}? {e}, {e, a, b, c},…

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Q: In the following Cayley table for a group G, C(2)= 1 2 4 5 3 4 4 2 3 4 2 3 4 1 3 4 2 4 5 {1,2,3,5}…

A: We Know that Let G be a group with operation *. The centre of G, denoted by C(G) is the subset of G…

Q: The following is a Cayley table for a group G. 2* 5*4 = 1 2 3 5 2 3 4 3 4 2 3 5 1 4 2 3 4 1 2 4 1.…

A: Cayley table for a group G is given as, The objective is to find 2*5*4 Since, G is a group. Hence,…

Q: Which of the following is nontrivial proper sub- group of Z4? {0, 2} O Diophantus of Alexandria {0,…

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Q: The following is a Cayley table for a group G. 2* 3 * 4 = 1 2 3 4 5 2 3 4 2 4 1 2 3 4 2 3 4 2 4 1 2…

A: From given table it is clear that

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: 2. Let A, B, C, and D be four groups such that A-C and B D. Show that A x B Cx D.

A: We will use the face that A isomorphic to C  and B isomorphic to D to prove the asked result.

Q: The following is a Cayley table for a group G. 2* 5 * 4 = * 1 2 3 4 5 1 2 3 4 5 2 3 4 5 2 3 4 5 2 4…

A: 1

Q: Compute the orders of the following groups.a. U(3), U(4), U(12)b. U(5), U(7), U(35)c. U(4), U(5),…

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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: Suppose that N and M are two normal subgroups of a group and that IOM = {e}. Show that for any n E…

A: Given: N and M are two normal subgroups of G and N ∩ M = {e} To prove: nm = mn for any n∈ N and m∈ M

Q: (b) Consider the incomplete character table for a group given below:

A: The table for the incomplete character is,   1 a b c d x1 1 1 1 1 1 x2 1 1 -1 -1 1 x3 1 1…

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: Which of the following groups are isomorphic? i) Z/75Z ii) Z/3Z ® Z/25Z iii) Z/15Z @ Z/5Z iv) Z/3Z @…

A: Groups (i) ℤ/75ℤ and (ii) ℤ/3ℤ⊕ℤ/25ℤ are isomorphic. Groups (iii) ℤ/15ℤ⊕ℤ/5ℤ and (iv) ℤ/3ℤ⊕ℤ/5ℤ⊕ℤ/5ℤ…

Q: (b) Complete the following character table of a group of order 12: 1 3 4 X1 X2 X3 4,

A: The character table of a group of order 12: