In the group Z, find а. (8, 14); b. (8, 13); с. (6, 15); d. (m, п); е. (12, 18, 45). In each part, find an integer k such that the subgroup is (k).

Q: Draw the S, Z and PI shaped membership functions to represent the young, middle-aged and old persons…

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Q: consider S3 and N = { (1), (12)} and H = { (1) , (123), (132)} both are subgroups of S3. Determine…

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Q: Let a be an element of order 8 in a group G. 6 Then order of a Select one: O a. 6 о Б.48 C. 2 d. 4

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Q: Find cyclic subgroups of S4 that have three different orders.

A: There are more subgroups than just the cyclic ones.   Trivial:  there is <e> = {e}.   There…

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: Give the order of the element in the factor group (2,3)+⟨(0,3)⟩ in (Z10×Z4)/⟨(0,3)⟩

A: The order of the element (2 3 +  <(0, 3)> in (Z⁠⁠⁠⁠⁠⁠⁠10×Z4) / <(0, 3)> is…

Q: What is the difference between a proper subgroup and a subgroup?

A: Here, we will tell the difference between a subgroup and a proper subgroup.Order of group means…

Q: Determine the groups of order 45

A: By Sylow's theorems, the number of Sylow 3-subgroups must divide 45 and be congruent to 1 modulo 3,…

Q: Identify the following Cayley digraph of a familiar group.

A: We have to identify the given Cayley digraph of familiar group.

Q: list the elements of the subgroup generated by the subet {2,3} of Z12

A: Let

Q: 2. Let G = Z50 and consider it's subgroup H = (5). Find all coset representatives of 3+ H.

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Q: Let H be a subgroup of G and let a, b E G. If aH = bH, then * На 1 — НЬ 1 О На %3Dнь На-1 %3D НЬ-1…

A: H is a subgroup of G and a,b belongs to G. If aH=bH then a and b lies in the same left coset of H.…

Q: In (Z,+) determine ({2,4,5}), the subgroup generated by the set {2,4,5). (As always, please provide…

A: In  , we have to determine , the subgroup generated by the set .We know thatDefinition : Given a…

Q: why are cosets not always subgroups? can you please give an example. thank you.

A: Cosets are not always subgroups because they do not necessarily satisfy all the requirements of a…

Q: How many elements does the group of rotations of a regular hexagon have?

A: The group of rotations of a regular hexagon is D6. And as the order of the group Dn is 2n.

Q: Find all the cosets of the subgroup 4Z of Z

A: In the group of integers under addition, the subgroup consists of all multiples of 4, i.e., .A coset…

Q: ow many subgroups of order 3 does D3 have? Write a number of subgroups only.

A: D3={1 , r , r2 , f , rf , r2f }  Or=3 Of=2    rmf=frn-m   n=3 , 0≤m≤3 O(rif)=2   0≤i≤2 3-order…

Q: Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} where a = (1 2 3 4) and b =…

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Q: How many subgroups of the group G = (Z30, +30)? а. 8 O b. 6 С. 5 O d. 7 O e. 4

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Q: 4. a) Let H be the set of elements [a b] of G of GL(2,R) such that ab- bc = 1. Show that H is a…

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Q: Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} %3D where a = (1 23 4) and b…

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Q: Discuss the mapping properties of for ny 2 -15 N

A: The mapping properties of z1n has to be discussed. Here the range is n≥2.

Q: How many Sylow 5-subgroups of S5 are there? Exhibit two.

A: We have to find Sylow 5-subgroups of S_5

Q: The elements of the quotient group (2/8). 12 (0,1,2,3,...,8) 1- 2- -208,-108,008,108,208,...) 3- 4-…

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Q: Find the subgroup of Dn generated by r2 and r2s, distinguishing carefully between the cases n odd…

A: Subgroup generated by r2 and r^2s in Dn​: We will analyze the subgroup generated by r2 and r2s in…

Q: b. Find the center and the commutator subgroup of S2 × Z7.

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