Q1)) prove or disprove ( 1. Let (G, *) be a group, if x*y = y*x then (x*y)" = x" * y". 2. Each cyclic group is a commutative group. 3. If (G, *) is a group then (a*b) = a+b* iff (G, *) is a commutative group.

Q: Prove that a group of order n greater than 2 cannot have a subgroupof order n – 1.

A: Given: To Prove: G cannot have a subgroup of order n-1.

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: 9. Prove that a group of order 3 must be cyclic.

A:

Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

A:

Q: Show that U5 andZ4 are isomorphic groups?

A: U(5)= {1,2,3,4}, <2> = {2, 22 = 4, 23 = 8, 24 = 1} = U(5) Therefore, U(5) is a cyclic group of…

Q: 4. If a is an element of order m in a group G and ak = e, prove that m divides k. %3D

A: Step:-1 Given that a is an element of order m in a group G and ak=e. As given o(a)=m then m is the…

Q: 3. Prove that (Z/7Z)* is a cyclic group by finding a generator.

A: Using trial and error method, seek for an element of order 6.

Q: Explain why a group of order 4m where m is odd must have a subgroupisomorphic to Z4 or Z2 ⊕ Z2 but…

A:

Q: Let G be a group with the property that for any x, y, z in the group,xy = zx implies y = z. Prove…

A:

Q: The symmetry group of a nonsquare rectangle is an Abelian groupof order 4. Is it isomorphic to Z4 or…

A:

Q: 7. If x and g are elements of group G, prove that x=g 'xg. Warning: You may not assume that G is…

A:

Q: does the set of polynomials with real coefficients of degree 5 specify a group under the addition of…

A:

Q: et G be a group with order n, with n > 2. Prove that G has an element of prime order.

A:

Q: List the six elements of GL(2, Z2). Show that this group is non-Abelian by finding two elements that…

A:

Q: (3) Show that 2Z is isomorphic to Z. Conclude that a group can be isomorphic to one of its proper…

A: (2ℤ , +) is  isomorphic to (ℤ , +) .              Define f :(ℤ , +) →(2ℤ , +) by…

Q: Show that ( Z,,+,) is a cyclic group generated by 3.

A:

Q: find Aut(Z30).  Use the FUndamental Theorem of Abelian Groups to express this group as an external…

A:

Q: 6. Embed the group Qs into the SU(2).

A: Given: Q0=e,i,j,k  e-2=e, i2=j2=k2=ijk=e, Where, e is the identity element and e commutes with the…

Q: 1- Prove that if (Q - {0},) is a group, and H = 1+2n 1+2m 9 n, m e Z} is a subset of Q-{0}, then…

A: We need to prove that H=1+2n1+2m∋n,m∈Z is a subgroup of Q-0 A subset W of a group V is said to be a…

Q: 2. Use one of the Subgroups Tests from Chapter 3 to prove that when G is an Abelian group and when n…

A:

Q: Prove that a cyclic group with even number of elements contains ex- actly one element of order 2.

A: The solution is given as

Q: (3) Suppose n= |T(x)| and d=|x| are both finite. Then, using fact 3 about powers in finite cyclic…

A:

Q: Prove that a group of even order must have an odd number of elementsof order 2.

A: Given: The statement, "a group of even order must have an odd number of elementsof order 2."

Q: Suppose that G is a finite group and that Z10 is a homomorphicimage of G. What can we say about |G|?…

A:

Q: determine whether the binary operation * defined by a*b=ab gives a group structures on Z. if it is…

A:

Q: 7. Prove that if G is a group of order 1045 and H€ Syl19 (G), K € Syl₁1 (G), then KG a and HC Z(G).…

A: As per policy, we are solving only the first Question, Please post multiple Questions separately.

Q: Suppose x is an element of a cyclic group of order 15 and x3 = x7 = x°. Determine |x13].

A: According to a theorem in group theory , If G is a finite group and a∈G be an element in the group…

Q: 4. Prove that the set H = nEZ is a cyclic subgroup of the group GL(2, R).

A:

Q: 4. Construct a 2-dimensional CW-complex whose fundamental group is Z x Z/2 (and prove it).

A: Please check the detailed sol" in next step

Q: (a) Compute the list of subgroups of the group Z/45Z and draw the lattice of subgroups. (prove that…

A: In the given question we have to write all the subgroup of the group ℤ45ℤ and also draw the the…

Q: not

A:

Q: Verify the corollary to the Fundamental Theorem of FiniteAbelian Groups in the case that the group…

A: To verify corollary to the Fundamental Theorem Of Finite Abelian Groups Where, G is a group of order…

Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

A:

Q: 2- Let (C,) be the group of non-zero -complex number and let H = {1,-1, i, -1}. Show that (H,;) is a…

A: We will be using definition of subgroup and verify that H indeed satisfy the definition.

Q: determine whether the binary  operation * defined by a*b=ab gives group structure of Z. if it is not…

A:

Q: Let G be a group of order 25. Prove G is cyclic or g^5=e for all g in G.  Generalize to any group of…

A: The Result to be proved is: If G is a group of order p2, where p is a prime, then either G is cyclic…

Q: 2) Determine whether or not the groups Z10 × Z4 and Z, × Z20 are isomorphic. Justify your answer.

A:

Q: 2- Let (C,) be the group of non-zero -complex number and let H = {1, –1, i, -1}. Show that (H,;) is…

A:

Q: Decide if the abelian group Z/2 × Z/2 is cyclic or not. Prove your answer

A:

Q: Show that group U(1) is isomorphic to group SO(2)

A: See the attachment.

Q: 4. Consider the additive group Z. Z Prove that nZ Zn for any neZ+.

A: We know that a group G is said to a cyclic group if there exists an element x of the group G such…

Q: Prove that a finite group is the union of proper subgroups if andonly if the group is not cyclic

A: union of proper subgroups  proof: Let G be a finite group. In the first place, we are going the…

Q: show that under complex multiplication, G={1,-1.i,-i} is an abelian group?

A: we have proved this by cayley table.

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: 15*. Find an explicit epimorphism from Z24 onto a group of order 6. (In your work, identify the…

A: To construct a homomorphism from Z24 , which is onto a group of order 6.

Q: Show that a group of order 12 cannot have nine elements of order 2.

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

Q: Suppose G is a group in which all nonidentity elements have order 2. Prove that G is abelian.

A:

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 0:Z50-Z15 be a group homomorphism with 0(x)=4x. Then, Ker(Ø)= {0, 10, 20, 30, 40)

A:

Types of Bias

Statistics is the study of data collection, organization, analysis, interpretation, and presentation. Statistical bias is a characteristic of a statistical technique or its findings in which the expected value deviates from the actual root quantitative parameter being estimated. According to the actual definition of bias, it refers to the tendency of a statistic to overestimate or underestimate a parameter.