The set G = {1, 5, 7, 11, 25, 29, 31, 35} is a group under multiplication modulo 48, with the group table below. X48 1 5 7 11 25 29 31 35 1 1 5 7 11 25 29 31 35 ст 5 5 25 35 7 29 1 11 31 7 7 35 1 29 31 11 25 5 11 11 7 29 25 35 31 5 1 25 25 29 31 35 1 5 7 11 29 29 1 11 31 5 25 35 7 31 31 11 25 5 7 35 1 29 35 35 31 5 1 11 7 29 25 (a) How many elements of order 2 does G have? (b) State a standard group isomorphic to G, briefly justifying your answer. (c) Write down all the distinct left cosets of the subgroup N = {1,7} in G. (d) Justify the statement that N is a normal subgroup of G. (e) Write down the group table of the quotient group G/N. (f) State a standard group isomorphic to G/N.

Q: a) Find all m 1 such that 27 = 9 (mod m). [1 0 1] b) If A = |0 1 11 1| is a zero-one matrix, find A…

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Q: c. Complete the following statements: The group Sg = Sym (Ng) of all permutations on the set Ng…

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Q: ication modulo 4 is not a

A: Show that {1, 2, 3} under multiplication modulo 4 is not a group but that {1, 2, 3, 4} under…

Q: list the elements of the subgroup generated by the subset {18, 24, 39} of Z

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Q: Let H be a subgroup of a group G and a, be G. Then bE aH if and only if * O a-1b eH O ab-1 eH O None…

A: We know that  b∈bH                    (1) We know that  aH = bH if and only if a-1b ∈H…

Q: Prove O3 is not a group

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Q: If p is a prime number the set {1,2,3,...,(p-1)}  is a group under multiplication modulo p.

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Q: Is {0, 2, 4, 6} a subgroup of Zg? Justify your answer.

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Q: Consider from the b the Symmetric group. 1 2 3 5 6 7 7 5 4 1 2 36 3 5 6 = 3 7 15 a) Express each of…

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Q: How many elements of a cyclic group with order 10 have order dividing 5?

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Q: The set G = {1, 9, 31, 39, 41, 49, 71, 79} is a group under multiplicati modulo 80, with the group…

A: ( U(80), .) is group of order 32. Here G is a subgroup of (U(80),.).

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

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Q: Show that no proper subgroup of S4 contains both (1,2,3, 4) and (1,2).

A: We have to show that no proper subgroup of S4 contains both 1,2,3,4 and 1,2. We know that the…

Q: X={1,2,3,4} .The permutation group S4=

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Q: Show that the set (5,10,25, 35} is a group under multiplication modulo 40 by constructing its Cayley…

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Q: The elements of the quotient group (2/8).) are {} 2- -208,-108,008,108,208,...) 3- (0,1,2,3,...,8)…

A: Given that the quotient group Z8.⊗. We have to find the elements of the quotient group Z8.⊗.…

Q: If a group has exactly 24 elements of order 6, how many cyclic subgroupsof order 6 does it have?

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Q: The character table of every point group contains more than one representation all of whose…

A: The statement is true.  Every point group has a trivial representation, which is a representation…

Q: Exercise 7.6. Show that each row and column of the group table contains all of the elements of G…

A: Group table and a finite group G.

Q: 3. Show that the following permutations: e = 12345 67 8 a = 12345 678, B = 1 2 3 4 5 678 Y =…

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Q: Find the index of in the group Z24

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Q: How many subgroups are there in Z/14Z?

A: Subgroup: Let (G, ∙) be a group and S be a non‐empty subset of G.Then, S is called a subgroup of G…

Q: Alte

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Q: Q/ One of the following subsets of z, is not a subgroup ? 32 1) [0,8,16,24] 2)…

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Q: 10. If 4,42..a, is an element in a group, what is (4,az..a,) = ?

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Q: Suppose (G, ∗), (H, ∗), and (K, ∗) are groups with H, K ⊂ G. Show that (H ∩ K, ∗) is a group

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Q: 2. Is the set Z3 = {0,1,2} form a group with respect to addition modulo 3 how about to…

A: We use caley's table to verify the properties of a group.