(1).* 4 and |6| = 2, Let G be a group. Let a, b EG such that |a| and a b = ba. Then determine |ba|, i.e., the order of the group element ba e G (2). It is known that 7 € U(20). Determine 7-1, |7| and |7-1|

Q: 4) Find the smallest integer n such that an+3 = a. 5) Prove that a is an odd permutation. 6) Let ß =…

A: Respected student, By Bartleby guidelines, we have solved your three subparts out of 4, please…

Q: Which of the following groups are solvable groups: Z/nZ, D8, and S4.

A:

Q: Which of the following groups are cyclic?a. U(35)b. U5(40)c. U8(40)

A: Given: (a) U35(b) U540(c) U840 We have to check which one of the given groups is cyclic.

Q: Suppose that |a| and |b| are elements of a group and a and b commute.If |a| = 5 and |b| = 16, prove…

A: Given, a and b are elements of a group such that a and b commute.Also, given |a| = 5 and |b| = 16.…

Q: a(H n K) Let G is a group, H, K C G, and a e G. Is it the case that aH N aK? Provide a proof or…

A: According to the given information,

Q: Build the operation table ∗ on G = {e, a, b} knowing that (G, ∗) is a group.

A:

Q: Determine whether the following is a semigroup, monoid, or a group. 1. G=Z, a*b = a + ab 2. G=2Z,…

A:

Q: Go Let G be a group and a, b = G. Prove that (ab)−¹ = b¯¹a-¹. Let G be a group. Prove that for all…

A:

Q: Let G be group, aeG ond lal=n. Show that laml = (m,n) %3D

A:

Q: In each case, find elements a and b from a group such that |a| = |b| = 2.   a. |ab| = 3 b. |ab| =…

A: a=2b=2

Q: Consider the group G=A4. List the elements of A4 in cycle notation, and determin the order of A4.

A: Consider the group G=A4.  We have to list the elements of A4 in cycle notation, and we have to…

Q: Is the set Z a group under the operation a * b = a + b – 1? Justify your answer.

A:

Q: Select the correct answer below: No, none of the four properties are satisfied. No, the closure and…

A:

Q: Show that (Q*,-) is not cyclic..

A: We use the basic definitions of cyclic groups to solve this problem. The detailed solution is…

Q: Let G be a group of odd order. Show that for all a E G there exists b E G such that a = b?.

A: Consider the given information, Let G be a group of odd order then, |G|=2k+1 where k belongs to…

Q: Let X, Y be cyclic groups of order 2, 3 respectively. Find the orders of the elements of XxY.

A: Let X, Y be cyclic groups of order 2,3 respectively then we have to find the   the orders of the…

Q: Let G be a group. Assume a and b are elements of G and set H₁ = and H₂ =. Assume |H₁| = 21 and |H₂|…

A:

Q: 6. Complete the multiplication table for the group G = {a, b, c, d} a d a b d

A: QuestionDetails: Part of the multiplication table forthe group G = {a,b,c,d} is given. Complete the…

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: In the frieze group F7, write x2yzxz in the form xnymz k.

A: It is given that, F7=xnymzk|n∈Z, m=0 or m=1, k=0 or k=1. we need to write x2yzxz  in the form of…

Q: Let G =<a, b | a2 =e, b2 =e, aba =bab>. To what familiar groupis G isomorphic?

A: Here given that  G =<a, b | a2 =e, b2 =e, aba =bab>. To what familiar group is G isomorphic…

Q: Let G = <a, b | a6 = b3 = e, b-1ab = a3>. How many elements doesG have? To what familiar group…

A: Given set is G = <a, b | a^6 = b^3 = e, b^-1ab = a^3>.

Q: Let G, H, and K be groups. Show that (G x H) × K = G × (H × K). Hint: An element of (G × H) × K will…

A:

Q: Let G = <a, b | a6 = b3 = e, b-1ab = a3>. How many elements doesG have? To what familiar group…

A:

Q: Let G = U(32) and H = {1, 15}. The group GIH is isomorphic to one of Zg, Z, O Z, or Z, O Z, O Z,.…

A:

Q: 6.2 Which of the following groups are cyclic? a) Z12×Z, b) Z10×Z85 c) Z4xZ25x Z, d) ZnxZ2 x Z65 -21

A:

Q: 9. In a group, prove that (a"')"' - a

A:

Q: A. Let G={a,b) and examine the properties of the operation defined by the operation with aᵒa=a, a…

A: o a b a a b b b a                                                     Let   G =  a , b…

Q: Consider the groups  and .

A: We have to consider the groups

Q: 4.3. Let a and b be elements of a group G. Prove that ab and ba have the same order.

A:

Q: 2. Let G = {a, b, c, d, e} with the operation * defined by the table below. Determine if the system…

A: Let G be a set and * be a binary operation then (G,*) is a group if  (G,*) has closure property.…

Q: Please help me answer it

A:

Q: Let G be a group of order 3 and identity e. If the other elements are a and b, use the Latin square…

A: G be a group of order 3 Consider G=e,a,bwhere e is the identity then the Cayley table is…

Q: 1. Let G and H be groups, and let g E G and he H. Find a formula for the order of the element (g, h)…

A:

Q: Let G be a group. Consider H = {h € G| hg = gh for all g e G}. Then which one of the following is…

A:

Q: 2.6. Let G be a group. Define an opposite group Gº with law of composition a * b as follows: The…

A:

Q: Find order of all the elements of the Group = {1,-1, i, -i}. The binary operator * is defined as a *…

A: 1 is identity . so order of 1 is equal to 1.

Q: 6.2 Which of the following groups are cyclic? a) Z12×Z, b) Z10x Zgs c) Z4xZ2sx Z, d) Z22XZ21 xZ6s

A:

Q: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({a +bv2 : a,…

A:

Q: 1. Prove or disprove that each of the cises 52 of Section 3.1 is cyclic. a. Z2 × Z3

A:

Q: 52. Let G be a group. Complete the following statement: |Inn(G)| = 1 if and only if

A: Let G be group. Complete the following statement: |InnG|=1 if and only if _________

Q: The following is a Cayley table for a group G. 2* 5*4 = 1 2 3 5 2 3 4 3 4 2 3 5 1 4 2 3 4 1 2 4 1.…

A: Cayley table for a group G is given as, The objective is to find 2*5*4 Since, G is a group. Hence,…