Given the set of integers S:= {1, 2, 3, 4, 5, 6}. With G = S6, then the pair (G, o forms a group called the permutation group with respect the operation composition o. Knowing that G = S6, define the set GT= {a G: a(t) = t for all t = T} If the set T = {1,2} CS, then the subgroup GT that leaves T elementwise invariant is given by GT= {(1), (34), (56)} O True O False

Q: Let G= How many elements does G have?  To what familiar group is G isomorphic?

A:

Q: 24. If a and b are distinct group elements, prove that either a² + b² or a³ + b³.

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: Let * be defined on R+ by letting a * b = Vab. Then (IR†, *) is a

A:

Q: (a) class(x)| = |G: N(x)| for all x € G. (b) If H is a subgroup of G, then the number of conjugates…

A: Given Information: is a finite group.Normalizer: , where .To Prove:a) .b) Number of conjugates of…

Q: 3. Define an operation on G = R\{0} x R as follows: (a, b) (c,d) = (ac, bc + d) for all (a, b),…

A: 3. Define an operation * on G=ℝ\{0} ×ℝ as follows: (a,b)*(c,d)=(ac, bc+d) for all (a,b), (c,d) ∈G…

Q: 5. Let B = {-1, -i, i, 1}; where i² = multiplication. -1. Show (B,*) is a group where * represents 4

A:

Q: X={1,2,3}. The permutation group S3=

A: Given,            X = { 1,2,3 }

Q: 13. Which of the following is a subgroup of (R+, *) where a*b = (ab)/2? (Q, *) A. B. (Z, *) C. (Q+,…

A: Given that                    ℝ+, * is a group             where…

Q: If any two members P and Q of a group satisfy the relation P = Q'PQ, then P and Q commute. Select…

A:

Q: Show that S5 does not contain a subgroup of order 40 or 30.

A: Let’s assume that the H is a subgroup of S5. So,

Q: Let x and y be two vertices of a Cayley digraph. Explain why twopaths from x to y in the digraph…

A: x and y are two vertices of a Cayley digraph. ai’s and bj’s are generators of the Cayley digraph

Q: Find all the Sylow 3-subgroups of S4.

A: We have to find all the Sylow 3-subgroups of S4.

Q: X={1,2,3,4} .The permutation group S4=

A:

Q: Exhibit a Sylow 2-subgroup of S4. Describe an isomorphism fromthis group to D4.

A: It is given that a Sylow 2-subgroup of S4. So, we have, S4=4!=24=3×23. let k be a Sylow 2-subgroup…

Q: Exercise 7.6. Show that each row and column of the group table contains all of the elements of G…

A: Group table and a finite group G.

Q: Let x and y be two vertices of a Cayley digraph. Explain why twopaths from x to y in the digraph…

A: It is given that x and y be two vertices of a Cayley digraph. We can write two paths as a1,a2,a3,…an…

Q: 5. Let H= [ (û ¿)I a,b,c EIRI, show that: (H,+) is a sub gromp of (M,+).

A:

Q: Show that S5 does not contain a subgroup of order 40 or 30.

A: Let H be a subgroup of S5. we see that, HA5=HA5H∩A5 It follows that, A5≤HA5H∩A5≤S5 Since, S5=2A5, so…

Q: d) G=(Q-(0}, ); H=Q+ e) G=(Zg, ); H=(0,2,4} f) G=the set of 2-tuples of real numbers (a,b) under…

A: In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of…

Q: Let G = Z/36Z. Find all subgroups of G, describe all containments between these subgroups, and draw…

A:

Q: is a subgroup of Z, of order 12 O 12 5, 3.

A: The explanation of the solution as follows:

Q: 3) Let G and H be two groups with e and e' as identities. Define P = G x H by P = G x H = { (g,h) |…

A:

Q: Consider the groups  and .

A: We have to consider the groups

Q: Exhibit a Sylow 2-subgroup of S4. Describe an isomorphism fromthis group to D4

A: Given: A Sylow 2-subgroup of S4To find: Describe an isomorphism from this group to D4.

Q: Let a be an element of a group. Suppose abs(a5) = 12. Show that abs(a) = 12 or abs(a) = 60.

A:

Q: 5. Let G be a group, and let HCG be a subgroup. Show that if HG, then there exist elements a, b, a',…

A: According to the question,

Q: Show that the elements of S9 which send the numbers 2, 5, 7 among themselves form a subgroup of S9.…

A: 1. **Closure**: If any two permutations send 2, 5, and 7 among themselves, their composition must…

Q: (a) Consider (3,6) and (2,4) as elements of Z7xZ7. Compute (3,6) - (2,4). (b) Consider (3,6) and…

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: 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: Let G = U56 and define two subgroups H₁ and H₂ of G by := H₁ := Calculate H₁ H₂|. {1, 5, 9, 13, 25,…

A:

Q: 5. Prove (using the subgroup condition rather than using the axioms) that the set a b H = {( 6 ) :…

A:

Q: Suppose (G, ∗), (H, ∗), and (K, ∗) are groups with H, K ⊂ G. Show that (H ∩ K, ∗) is a group

A:

Q: for each of the group Z2,Z3,Z4,Z10 indicate which are cyclic. for those that are cyclic list all the…

A:

Q: Draw the lattice of the subgroups Z/20Z.

A:

Q: Let a,b be two elements in a group. Suppose a, b are of the same er. Let m be the smallest positive…

A:

Q: Let G {0, 1, 2, 3, 4, 5, 6, 7} and assume that G is a group with the operation satisfying the…

A: Introduction: A group is a collection of elements that are joined together by an associative…

Q: Let G be a finite group of order 4 containing no element of order 4. Explain why every nonidentity…

A: