Problem 1.0.3. Let a e G then we define the cyclic subgroup generated by a to be < a >:= {a"|n E Z} Some comments regarding the definition: aº = e where e is the identity element of the group. For n > 0, a" = a ★ a * · ·. * a (n-times) where * is the multiplication/operation of the group. ... Also note that n E Z so a-1 e< a > and for n < 0 we define a" (|n|-times). So for example a-3 = a-1 ★ a-1 ★a-1. (a-1) * ·.. ★ a-1 a) Let (G, *) = (Z,+) (integers with respect to addition) describe the elements of <1>, what is < 3 >? b) Let (G, *) = (Zs,+s) (residue classes modulo 8 with respect to the addition modulo 8). Find all such [a] such that < [a] >= Zs. This set of [a]'s should look familiar from somehwere..

Q: 1. Let M2(R) = {( ) la, b, c, d eR and ad bc + 0}. You already know that M2(R) forms a group under…

A: We have to prove that H is subgroup of M2(R) :

Q: An is the alternating group on n letters. What is |An|?

A:

Q: S[y] given by y = cosh æ + B sinhx, 0 0,

A:

Q: (h, k) E H x K. What is the order Let H and K be finite groups and let G = H x K. Let g of g?

A: To calculate the order of the element (h,k) in the direct product of groups

Q: 3. Let (G, *) be a group and let H and K be subgroups of G. Prove or disprove each of the following…

A: Given - Let G, * be a group and let H and K be subgroups of G. To find - Prove or disprove each of…

Q: 10. Let G be a group and H a subgroup of G. Let CH) - {gEG|gh = hg. VhEH}, Show that C(H) <G.

A:

Q: 2. Let G be the group of 2 x 2 matrices under addition and H = {(ab): a +d=0}. Prove that H is a…

A:

Q: 4. Suppose H is a normal subgroup of a finite group G. a. Explain why the order of the factor group…

A: Solution:-

Q: 6.10 Find all subgroups of Z,XZ4. 6.11 Find all subgroups of Z,xZ,>

A: To find  All subgroups of z2*z4

Q: Theorem 3.34: Let (G, *) be a group, then (1) {e}AG. (2) GAG (3) Cent(G) AG. Proof: (H.W.)

A:

Q: 2(a) 9. Let G be a group. Show that Z(G) =

A: we know that ZG of a group G is the subset of elements of G that commute with every element of G.…

Q: (4) Let (G, *) be a group and assume that z, a, b € G. Let also that c = x *a*r¹ and d = x * b*x-¹.…

A: Let  be  a  group  and  assume  that Suppose  that Since,  and  is  a  group, it follows  that…

Q: Updated the introduction of the problem. It was messed up. Introduction of the problem: Let G = D6…

A: To find the order of LR60 in G, we need to compute (LR60)^n for different values of n until we get…

Q: 3. Consider o = (3456) € S, and let H be the subgroup of S, generated by o. Justify all of your…

A: .

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

A: The given problem is related with group theory. G is a group and a ∈ G. Here, we write Cga instead…

Q: 16. Let G be a group and let g and h be in G. Show that . - as defined in #15.

A: Given: G is a group, g,h∈G, ϕg:G→G is a function defined as ϕg(x)=gxg-1 To prove: ϕg∘ϕh=ϕgh Let x∈G…

Q: 1.e. Consider the dihedral group (D3, ◦), ie. the symmetry group of an equilateral triangle.…

A: Symmetry group of an equilateral triangle have subgroup of order 2 and it is isomorphic to Z2

Q: Prove that G = {a+b√2: a, b € Q and a and b are not both zero} is a subgroup of R* under the group…

A: Given- G=a+b2: a,b∈ℚ and a and b are not both zero To prove- G is a subgroup of ℝ* under the group…

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)