G let then. [b, a]= be an group and Ta %3D

Q: G be and Let then O [a, b'] - [b, a]o ☺ [ai!, b] = [b, a ]a group a, bE G %3D whele xa a xa

A:

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: (a, b | a group of degree 3. Let G 10.1.2. Let Dg b? = e, ba a3b), and let S3 be the symmetric (b) ×…

A:

Q: 4. Recall that Z(G) = {r € G| gr = rg, Vg E G}. Show that Z(G) is a normal subgroup of G.

A:

Q: 10. Let E = Q(V2, V5). What is the order of the group Gal(E/Q)? What is the order of Gal(Q(V10/Q)?

A:

Q: be a group and Ha normal subgroup of G. Show that if x,y EG such that xyEH then yxEH Let G

A: Given: Let G be a group and H a normal subgroup of G.To show that x,y∈G suchthat xy∈H then yx∈H

Q: Prove that E(n) = {(A, ¤) : A e O(n) and E R"} is a group. %3D

A: Consider the given: E(n)={(A,x)} where A∈O(n)and x∈ℝn

Q: In the group D4, let a = sp and b = sp3. The inverse of of ab is: a) sp2 b) ps c) p? d) None a C

A:

Q: Let G be the group with presentation (x,y: 1² = 1, r² = y²₁ xy = y ¹). Decide how many elements are…

A: Consider the given group G with presentation, x,y:x4=1, x2=y2, xy=yx-1 it can be observed that it is…

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|

A:

Q: Let X be a group and we then let x and y be an element of X.  Prove that (x*y)^ -1 = a^-1 * b^-1 iff…

A: Since there are some mistakes in given typed question.question may like "Let X be a group and let…

Q: Let a and b belong to a group. If la| = 12, \b| = 22, and (a) N (b) + {e}, prove that a6 = bl1.

A:

Q: Is the set {3m + v3ni|m, n E Z, b|m – n} the normal subgroup of the (C, +)group?

A: given :

Q: 10. Given the group (S = {a,b,c, d},®) a a d ic a b a a d d The inverse of an element r, x E Sis D-…

A: Inverse element is an element which when operated with its inverse gives the identity

Q: W6 Assume that H, k, and k are SubgrouPs of the group G and k, , Ka 4 G. if HA k, = HN k Prove that…

A:

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: Let a and b belong to a group. If la] = 24 and |b| = 10, what are the possibilities for (a) n (b)i?

A:

Q: Let G be a group and let a e G. In the special case when A= {a},we write Cda) instead of CG({a}) for…

A: Consider the provided question, According to you we have to solve only question (3). (3)

Q: Let G = (Z,, +6) is an Abelian group then how many self - invertible elements in G? (A) 1 (B) 2 (C)…

A: To solve this problem, we use the defination of group.

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)?

A:

Q: 2. Let G be a group. Show that Z(G) = NEG CG(x).

A: Let G be a group. We know Z(G) denotes the center of the group G, CG(x) denotes the centralizer of x…

Q: Find normalizer and centralizer of the subset H = {1, r2 , r2 s} of the group D5= The dihedral group…

A: It is given that,  H = 1, r2, r2s be a subgroup of the dihedral group D5 We have to find the…

Q: If H and K are subgroups of G, |H|= 18 and |Kl=30 then a possible value of |HNK| is O18 8. O 4

A:

Q: Consider the group D4

A: Given: Group D4=a,b=e=1,a,a2,a3,b,ab,a2b,a3b and a=1 2 3 4, b= 2, 4 To find : The value of…

Q: Q\ Let (G,+) be a group such that G={(a,b): a,b ER}. Is ({(0,a): aER} ,+) sub group of (G,+).

A:

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

A:

Q: Show that (ℤ,∗)?ℎ??? r ∗s = (r +s)−(r ∙ s)??? ??? r,s ∈ ℤ is group using variable r, s and t.

A:

Q: Exercise 2: Let G be a group and a EG. For any m, neZ, prove that am*a = a"a" and (a" y" = am".

A:

Q: Q/ Let G = Z and a * b = a + b show that (G,* ) is a group or not.

A:

Q: 3. Suppose G = {e, v, w, x, y, z} is a group defined by the table below. %3D e W y e e W V V. W ey W…

A:

Q: Consider the subgroup H = (i) of the the group (C\{0},.). LIT TT TT?

A:

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

A:

Q: b' e GL(2, IR) а Is Ga subgroup of GL(2, IR)? Let G

A: Note that, the general linear group is

Q: 3.) In D4, the centralizer of the group at H is equal to? C(D) C(R90) A В C(D') C(V) D

A: Use the definition of D4.

Q: (c) The set K ={a'b' | 0 < i < |a| and 0 < j < |b| } is a group. Construct the complete group table…

A:

Q: -{8 a|a, b, d ER, ad # 0 }. Is Ha normal subgroup of GL(2, R)? Let H

A:

Q: Given the group (S = {a,b,c, d},8) %3D a d a b a b d - d The inverse of an element z, z e Sis a a с…

A: The solution is given as

Q: Find the solution set for each of the following with the representation of the group on the number…

A: To find - Find the solution set for each of the following with the representation of the group on…

Q: Let G be a group and a E G be such that o(r) 20. Find the orders of r, xt, a0, al4 and r30. %3D

A:

Q: Let be a group and Ha normal subgroup of G. Show that if y.VEG such that xyEH then yx EH

A:

Q: Let X = Cay(Z12, {±2, 6}) and Y = Cay(D12, {P, p°, T}), where D12 is dihedral group of order 12,…

A: Cayley graph or Cayley diagram is used to study the abstract nature of a group by drawing graphs…

Q: List the elements of the quotient groups of (a) (4Z, +) in (Z, +) (b) Z30/(6) (c) Z30/(J/H), where J…

A: Quotient group GH ={ Ha | a∈ G} where H is normal subgroup of group G. Here al given groups are…

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…

A:

Q: Which of the following ?groups is not cyclic GL(2, R) under addition componentwise. G = {a+b/2: a. b…

A: GL(2,R) under addition componentwise GL(2,R)=A| A≠0  For cyclic group there exists a matrix A such…

Q: Let G be a group. Let x, y e G be such that O(x) = 7, O(y) = 2, x^6 y = yx. Then O(xy) is O Infinity…

A:

Q: / Let x be anan- empty set, show that whe ther (P(X), n) is a a group of not

A:

Q: 25: Let R? = R × R = {(a, b) : a e R, be R} and T: R? → R² s.t. Ta b)(x, y) = (x + a, y + b) (a,.…

A: