6. Let G be GL(2, R), the general linear group of order 2 over R under multiplication. List the elements of the subgroup (A) of G for the given A, and give |(A)\. [. ]-v а. А в b. A =

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: Prove that a simple group of order 60 has a subgroup of order 6 anda subgroup of order 10.

A: If G is the simple group of order 60 That is | G | =60. |G| = 22 (3)(5). By using theorem, For every…

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: Show that SL(n, R) is a normal subgroup of GL(n, R). Further, by apply- ing Fundamental Theorem of…

A: Suppose, ϕ:GLn,R→R\0 such that ϕA=A for all A∈GLn,R Now, sinceA∈GLn,R if and only if A≠0 Now, we see…

Q: Prove that a group of order 7is cyclic.

A: Solution:-

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: 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: The group generated by the cycle (1,2) is a normal subgroup of the symmetric group S3. True or…

A: Given, the symmetric group S3={I, (12),(23),(13),(123),(132)}. The group generated by the cycle (12)…

Q: Show that the center of a group of order 60 cannot have order 4.

A:

Q: Give an example, with justification, of an abelian group of rank 7 and with torsion group being…

A: consider the equation

Q: 14*. Find an explicit epimorphism from S4 onto a group of order 4. (In your work, identify the image…

A: A mapping f from G=S4 to G’ group of order 4 is called homomorphism if :

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

A:

Q: Which of the following cannot be an order of a subgroup of Z12? 12, 3, 0, 4?

A: Since 0 does not divides 12.

Q: 9. Prove that if G is a group of order 60 with no non-trivial normal subgroups, then G has no…

A:

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

A:

Q: 1. Show that H={[0], [2], [4]} is a subgroup of a group (Z6+6). Obtain all the distinct left cosets…

A: Given that H=0,2,4 and let G=ℤ6,+6.

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

A:

Q: KE Syl-(G). Prove that (a). HG and KG. (U). G has a cyclic subgroup of order 77. Syl(G),

A:

Q: Describe all the elements in the cyclic subgroup of GL(2,R) generated by the given 2 × 2 matrix. -1

A: Let A=0-1-10∈GL2, R Then the cyclic subgroup generated by A denoted by A=An| n∈Z

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: 64. Express Ug(72)and U4(300)as an external direct product of cyclic groups of the form Zp

A: see my attachments

Q: Compute the center of generalized linear group for n=4

A: To find - Compute the center of generalized linear group for n=4

Q: Sz,0) be a permutation group. Then all elements in One to one, onto function. Onto function.

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: Show that every abelian group of order 255 (3)(5)(17) is isomorphic to Z55 and hence cyclic. [Ilint:…

A: We have to solve given problem:

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: At now how many elements can be contained in a cyclic subgroup of ?A

A: There will be exactly 9 elements in a cyclic subgroup of order 9.

Q: 2. Prove that a free group of rank > 1 has trivial center.

A: Given:Prove that a free group of rank>1 has trivial center

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: Determine the class equation for non-Abelian groups of orders 39and 55.

A: We have to determine the class equation for non-Abelian groups of orders 39 and 55.

Q: Q3: (A) Prove that 1. There is no simple group of order 200. 2. Every group of index 2 is normal.

A: Sol1:- Let G be a group of order 200  i.e O(G) = 200                = 5² × 8. G contains k Sylows…

Q: (i). There is a simple group of order 2021.

A:

Q: (7) Define GL2 (R) to be the group of invertible 2 x 2 matric manifold, cc this group has the…

A: Define GL2R  to be the group of invertible 2×2 matrices.  To prove that this group has the structure…

Q: Prove that every finite Abelian group can be expressed as the (external) direct product of cyclic…

A: Fundamental Theorem of Finite Abelian Groups: Every finite Abelian group is a direct product of…

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: Show that there are two Abelian groups of order 108 that haveexactly one subgroup of order 3.

A:

Q: not

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: 300Can someone please help me understand the following problem. I need to know how to start the…

A: G is the abelion group of order 16. It is isomorphic to,

Q: 8. Prove that if G is a group of order 60, then either G has 4 elements of order 5, or G has 24…

A: As per the policy, we are allowed to answer only one question at a time. So, I am answering second…

Q: If R2 is the plane considered as an (additive) abelian group, show that any line L through the L in…

A:

Q: (a) Let G be a non-cyclic group of order 121. How many subgroups does G have? Why? (b) Can you…

A:

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

A: we have proved this by cayley table.

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: 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: a. Show that (Q\{0}, + ) is an abelian (commutative) group where is defined as a•b= ab b. Find all…

A:

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

A:

Q: (4) Find the Galois group of the polynomial r + 1.

A: Since you have asked multiple question, we will solve any one question for you. If you want any…