5. A group G has subgroups {a,b},{a,6, c, d, f } and {a,d}. The identity element of G is А. d В. с С. Ь

Q: In (Z12, +12) , H, = {0,3,6,9} and H2 = {0,3} are tow subgroups of the group (Z12, +12), but (H, U…

A: We have given   H1=0,3,6,9 and H2=0,3 are two subgroups of the group Z12, +12. We can see here…

Q: (Z, +) is a group and infinite group

A: Let a binary operation '*' defined on a set G, then it forms a group (G,*) if it holds the following…

Q: 8. Show that (Z,,x) is a monoid. Is (Z,,x4) an abelian group? Justify your answer

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Q: 7. Show that 4 is a subgroup of S,

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Q: Give an example of a group of order 12 that has more than one subgroupof order 6.

A: Consider the group as follows, The order of a group is,

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

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Q: (H,*) is called a of (G,*) if (H,*) is a group.

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Q: shiw that in a group if x has an inverse y abd a right inverse r, then y and r are the same element

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Q: Suppose H and K are subgroups of a group G. If |H| = 12 and|K| = 35, find |H ⋂ K|. Generalize.

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Q: 5. If H = 122Z and K = 8Z are subgroups of (Z, +). Then H + K = ... %3D

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Q: 2. Let G be a group. Pro-

A: Let G be a group .

Q: 1. Determine all subgroups of the group (U13, ·)

A: The sub group of U13 is to be determined.

Q: 3. Consider the group (Z,*) where a * b = a + b – 1. Is this group cyclic?

A: 3. Given the group ℤ,* where a*b=a+b-1. Then, 1*x=x*1=x+1-1=x Here 1 serves as the identity for Z.

Q: 27. If g and h have orders 15 and 16 respectively in a group G, what is the order of (9) n (h)?

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Q: If a1, a2, . . . , an belong to a group, what is the inverse of a1a2 . . . an?

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Q: Let α,β ESs ( a = (1,8,5,7)(2,4) and B= (1,3,2,5,8,4,7,6). Compute aß. Symmetric group) where

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Q: Suppose G is a group and r, be G so that r = b and r = b. Solve for a in terms of b.

A: Given: G is a group, and x,b∈G, so that x3=b5 and x8=b2. Formula used: Basic formula in power and…

Q: In (Z10, +10) the cyclic subgroup generated by 2 is (0,2,4,6,8). True False If G = {-i,i,-1,1} be a…

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Q: If H and K are normal subgroups of G, show that their intersection is also a normal subgroup. To do…

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Q: 46. Determine whether (Z, - {0},6 ) is it a group or not? Explain your answer?

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Q: c and of d be elements of group G such that the order of c is 5 and the order of d is 3 respec-…

A: #Dear user there is a mistake in the question the assumption is for the element c and d of a group…

Q: 4-Let (Z12, +12) be a group and let S={4,6}, find subgroup H generated by S. if exist

A: I have used the definition of subgroup generated by a subset.

Q: (b) Suppose G is a group, H, K < G, |H|= 30, |K| = 20, and |HN K| = 10. What is |HK|? %3D %3D

A: By theorem, Order of product of two subgroup of finite order       Let the two subgroups be H and K…

Q: 2- Find the cyclic subgroups in (Z,, +,) generated by 3,5.

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Q: Determine the order of (Z ⨁ Z)/&lt;(2, 2)&gt;. Is the group cyclic?

A: Given, the group                 We have to find the order of the group and also check, This is a…

Q: If G is an infinite group, what can you say about the number ofelements of order 8 in the group?…

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Q: Let Ø:Z50→Z15 be a group homomorphism with Ø(x)=7x. Then, Ker(Ø)= * O {0, 10, 20, 30, 40} None of…

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Q: 4

A: To identify the required cyclic subgroups in the given groups

Q: 4. Which of the groups U(14), Z6, S3 are isomorphic?

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Q: Q1/ If (H,*) is collection of subgroups of (G,*) then (U H,*) is subgroup of (G,*)

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Q: The group (Z6,6) contains only 4 subgroups

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Q: Let (Z12, +12) be a group , if we take {0,4,8} for the set H then ({0,4,6}, +12) is evidently a…

A: Let H=0, 4, 6 We know that the operation in ℤ12 is addition. So, the element of left coset is of the…

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.

Q: 2. What is the order of the element 32 in the group Z36?

A: Modular groups  are cyclic groups. A group G is cyclic if G=&lt;g&gt; for some g in G, where…

Q: Show that if aEG, where G is a group and |a| = n then : %3D a' = a' if and only if n divides. -j

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Q: 2. Show that in a group if x has inverse y and y and a right an inverse r, then y and r are the same…

A: We need to show that in group if x has an inverse y and a right inverse r, then y and r the same…

Q: {1,2, 3} under multiplication modulo 4 is not a group. {1,2, 3, 4} under multiplication modulo 5 is…

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Q: 46. Determine whether (Z, - {0}, 6 ) is it a group or not? Explain your answer?

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Q: Consider S4 and its subgroups H = {i,(12)(34),(13)(24),(14)(23)} and K = {i,(123),(132)}. For a =…

A: Note: We are using the simple procedure that is by direct calculation. We are given the group S4 and…

Q: • 2. List the elements of the subgroups (1) , (2), (3),(4), (6) and (0) in Z12.

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Q: The group (Z6,+6) contains only 4 subgroups

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Q: Suppose that X and Y are subgroups of G if |X|=28 and |Y|=42, then what is

A: "According to Bartleby Guideline, Handwritten solution are not provided" Given,            |x|=28…

Q: Decide whether (Z, -) forms a group where : Z xZ Z (a) is the usual operation of subtraction, i.e.…

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: Consider the set of permutations V = {(1), (1 2) (3 4), (1 3) (2 4), (1 4) (2 3)}. Determine whether…

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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: 8. Show that (Z,,×s) is a monoid. Is (Z.,×6) an abelian group? Justify your answer

A: Note: since you have posted multiple questions . As per our guidelines we are supposed to solve one…

Q: 1- (Z,+) is a subgroup of (Q,+) 2- (Q,+) is a a subgroup of (R, +) 3- (R,+) is a a subgroup of (C,+)…

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Q: Suppose that a group G has order 8, but is not cyclic. Then that g for all g e G Select one: а. 1 3…

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Q: Which of the following is nontrivial proper sub-group of Z4? {0,2}, {0,3}, {0,1}

A: We have given a group Z4 ,  and we have to find which of the following sets {0,2},{0,3},{0,1} is a…

Q: a) How many subgroups does (Z10,O) have? What are they? b) How many subgroups does (Z74,Ð) have?…

A: 2. a) Consider the group ℤ10, ⊕. The elements of the above group are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9…