6. Let H be a subset of a group G that satisfies the following properties a. H + 0 b. if x e H andy e H, xy e H. C. if x e H then x- e H. Prove e e H.
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: - Show that the following subset is a subgroup. H = {o e S, l0(n) = n} S,
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Q: The following is a Cayley table for a group G, 2 * 3 * 4 = 3 1 2. 4 主 3. 4 2 1 21 4 345
A: For group, 2*3*4=(2*3)*4.
Q: If g and h are elements from a group, prove that ΦgΦh = Φgh
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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: Define ∗ on ℚ+ by a ∗ b =ab/2 . Show that ⟨ℚ+,∗⟩ is a group.
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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: Is the set Z a group under the operation a * b = a + b – 1? Justify your answer.
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Q: Let G be a group and suppose that a * b * c = e. Show that b * c *a = e.
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Q: Let G be a group and let H< G. If [G: H] = 16 and |H| = 21, then what is |G|?
A: The expression, G:H can be written as GH .
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: Consider the group (Z,*) defined as a*b=a+b , then identity (Neutral) element is a 1 b -1…
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Q: Z, x is not a group.
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Q: Given that A and B is a group. Find out if : A→B is a homomorphism. If it is a homomorphism, also…
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Q: G let then. [b, a]= be an group and Ta %3D
A: Given that G is a group and also a,b,c∈G. To prove that b,a= a,b-1 Since G is a group, it satisfies…
Q: Let a and b be elements of a group. If |a| and |b| are relatively prime, show that intersects =…
A: Let m and n be the order of the elements a and b of a group G. Given that the orders of a and b are…
Q: Let a and b belong to a group. If |a| = 10 and |b| = 21, show that n = {e}
A: Consider a group G. Let a and b be elements of the group G such that a=10 and b=21. Consider the…
Q: 1. Assume (X, o) and (Y, on X x Y as are groups. Let X × Y = {(x, y) |x € X,y E Y} and define the…
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Q: 10, Let (G, *) be a group and a, b, c E G. (a) Prove if a *b = a * c, then 6 = C (b) Prove if b * a…
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Q: What are the three things we need to show to prove that an ordered pair is a group?
A: We have to give the properties of an ordered pair to prove that it is a group.
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: 5) In each of parts (a) to (c) show that for specified group G and subgroup A of G, Cg(A) = A and…
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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: Can you write a group homomorphism as φ (gh) as φ(hg)? Are they the same thing?
A: The given homomorphism ϕgh, ϕhg The objective is to find whether the ϕgh,ϕhg are same.
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: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({3* : k E…
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Q: belong to a group. If |a| = 12, |b| = 22, and (a) N (b) # {e}, prove that a® = b'1.
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Q: Prove or disprove, as appropriate: If G x H is a cyclic group then G and H are cyclic groups.
A: GIven two groups G and H such that GxH is cyclic. True or false: G and H themselves are cyclic
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 G be a defimed by is a group action. group and A= G. Show a*X = ANA' , aXE G that *
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Q: Let G be a group and let a, be G such that la = n and 6| = m. Suppose (a) n (b) = (ea). Prove that…
A: According to the given information, let G be a group.
Q: 6. If G is a group and a is an element of G, show that C(a) = C(a')
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Q: 1. State, with reasons, which of the following statements are true and which are false. (a) The…
A: Given Data: (a) The dihedral group D6 has exactly six subgroups of order 2. (b) If F is a free group…
Q: Assume (X,o) and (Y, on X x Y as are groups. Let X × Y = {(x, y) | æ € X, y E Y} and define the…
A: The given question is related to group theory. Given: X , ∘ and Y , ∙ are groups. Let X × Y = x,y |…
Q: Q/ Let G = Z and a * b = a + b show that (G,* ) is a group or not.
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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: Is the set Z a group under the operation a * b = a – b + ab? Justify your answer.
A: Check the associative property. Take a = 2, b = 3 and c =4. (a*b)*c = (2*3)*4 =…
Q: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian
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Q: any two elements x and y in a group G, there is a unique element z in G such that y = xz.
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Q: G, ba = ca implies b = c and ab = ac implies b = c for elements a, b, c E G. 31. Show that if a? = e…
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Q: .A group (M,*) is said to be abelian if إختر أحد الخيارات (x+y)=(y+x) .a O (y*x)=(x+y) .b O…
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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: 64
A: Under the given conditions, to show that the cyclic groups generated by a and b have only common…
Q: F. Let a e G where G is a group. What shall you show to prove that a= q?
A: Solution: Given G is a group and a∈G is an element. Here a-1=q
Q: and are groups. Let (2, 9)| E A, y E. and denne the operation on X x Y as (T1, 41) * (x2, Y2) = (x1…
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Q: 7. Prove that if G is a group of order 1045 and H€ Syl₁9 (G), K € Syl (G), then KG and HC Z(G).
A: 7) Let G be a group of order 1045 and H∈Syl19(G) , K∈Syl11(G). To show: K⊲G and H⊆Z(G). As per…
Q: / Let x be anan- empty set, show that whe ther (P(X), n) is a a group of not
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Q: The set numbers Q and R under addition is a cyclic group. True or False then why
A: Solution
Q: Let X be a non-empty set, Show that whether (P(X),n) is a group or not
A: given Let x be a non empty set, Show that whether (p(x),∩) is a group or not we need to show it is…
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- (See Exercise 31.) Suppose G is a group that is transitive on 1,2,...,n, and let ki be the subgroup that leaves each of the elements 1,2,...,i fixed: Ki=gGg(k)=kfork=1,2,...,i For i=1,2,...,n. Prove that G=Sn if and only if HiHj for all pairs i,j such that ij and in1. A subgroup H of the group Sn is called transitive on B=1,2,....,n if for each pair i,j of elements of B there exists an element hH such that h(i)=j. Suppose G is a group that is transitive on 1,2,....,n, and let Hi be the subgroup of G that leaves i fixed: Hi=gGg(i)=i For i=1,2,...,n. Prove that G=nHi.42. For an arbitrary set , the power set was defined in Section by , and addition in was defined by Prove that is a group with respect to this operation of addition. If has distinct elements, state the order of .12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.
- In Exercises and, the given table defines an operation of multiplication on the set. In each case, find a condition in Definition that fails to hold, and thereby show that is not a group. 15. See Figure.Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?In Exercises 15 and 16, the given table defines an operation of multiplication on the set S={ e,a,b,c }. In each case, find a condition in Definition 3.1 that fails to hold, and thereby show that S is not a group. See Figure 3.7 e a b c e e a b c a e a b c b e a b c c e a b c
- Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.Exercises 13. For each of the following values of, find all subgroups of the group described in Exercise, addition and state their order. a. b. c. d. e. f.Label each of the following statements as either true or false. The Generalized Associative Law applies to any group, no matter what the group operation is.
- If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.In Exercises 3 and 4, let G be the octic group D4=e,,2,3,,,, in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group D4 of rigid motions of a square The elements of the group D4 are as follows: 1. the identity mapping e=(1) 2. the counterclockwise rotation =(1,2,3,4) through 900 about the center O 3. the counterclockwise rotation 2=(1,3)(2,4) through 1800 about the center O 4. the counterclockwise rotation 3=(1,4,3,2) through 2700 about the center O 5. the reflection =(1,4)(2,3) about the horizontal line h 6. the reflection =(2,4) about the diagonal d1 7. the reflection =(1,2)(3,4) about the vertical line v 8. the reflection =(1,3) about the diagonal d2. The dihedral group D4=e,,2,3,,,, of rigid motions of the square is also known as the octic group. The multiplication table for D4 is requested in Exercise 20 of this section.In Exercises 3 and 4, let be the octic group in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let be the subgroup of the octic group . Find the distinct left cosets of in , write out their elements, partition into left cosets of , and give . Find the distinct right cosets of in , write out their elements, and partition into right cosets of . Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group of rigid motions of a square The elements of the group are as follows: 1. the identity mapping 2. the counterclockwise rotation through about the center 3. the counterclockwise rotation through about the center 4. the counterclockwise rotation through about the center 5. the reflection about the horizontal line 6. the reflection about the diagonal 7. the reflection about the vertical line 8. the reflection about the diagonal . The dihedral group of rigid motions of the square is also known as the octic group. The multiplication table for is requested in Exercise 20 of this section.