G E E LE B D None All (ii), (iii) (i), (iii) (i), (ii) Let G be a finite group of order 100. Which of the following are correct? (i): There is an element a E G having an order 40. (ii): If a G, then a¹00 = e. (iii): There are no subgroups of order 80. E G

Q: (ii) Let K={(), (1 2), (3 4), (5 6), (1 2)(34), (1 2) (5 6), (3 4) (5 6), (1 2)(3 4)(5 6)} with the…

A:

Q: Prove that a group of order 28 must have an element of order 2.

A:

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: -) Write the Cayley table for the group U(15).

A: U(15) is the set of all elements in ℤ15={0,1,2,3,4,5,6,7,8,9,10,11,12,13,14} ; x≡x ,such that…

Q: 2n 5) Let G be a group such that |G| = (xd,)!, and |H|= (n – 1), where H a Subgroup of G, then- n(n…

A:

Q: To Let G = 31, 2, 3, 4, 5, 63. Given that (6₂.) is a group with multiplication table below? T 1 22…

A:

Q: 2. Let G be a group and H1, H2 <G subgroups. (a) Suppose |H1| = 12 and |H2 = 28, prowe H1 n H2 is…

A:

Q: 2. Let G be a group. Prove or disprove that z= { XE G: xg = gx for all ge G} is a Subgroup of G.

A:

Q: Exercise 5: (a) Show that permutations (12)(34), (13)(24). (14)(23), and the identical permutation e…

A:

Q: Exercises 3(a) Prove that the groups $ and D6 are isomophic ; (e) Preve that the additive and the…

A:

Q: 5. Let G be a group. (a) Prove that Inn(G) is a subgroup of Aut (G) (b) Prove that for any € Aut(G)…

A:

Q: (b) Let G be a group and let x E G. Suppose that (x) is a cyclic group of order N. Let m and n be…

A: We make use of the gcd to answer this question. DEtails are given in step 2.

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: (a) Let G = {e, a, . .. , aº | a10 = e} be the cyclic group of order 10. For which m, is G = (am)?…

A: As per our guideline we are supposed to answer only first asked  question. Kindly repost other…

Q: O O O O If G is a finit subgroups c order of G is 80 60 90- 30

A:

Q: Let G be a group and a e G. Show that o(a) = o(a-). order n, then ba also has order n.

A:

Q: The class equation for a group G can be written |G| = |Z(G)| + 2|CI(a;)|, where the sum runs over…

A:

Q: Depine Dihedral group of order 8. Write two non-trival Subgroups of Dihedral group

A: Dihedral group of order 8 is given by     D4 = group of symmetries of a square          = {R0, R90,…

Q: I denotes the set of real numbers and (*) is an operation on R such that a * B = a +B+ aß for ali a,…

A:

Q: Which of the following statements about the order of elements in the group GL2(R) is not true? (a)…

A: Ans: (d) not true.For details please see the explanation.If you have any problem then let me know in…

Q: 1. Let (G, *) be a finite group of order n. Then (a). Let n = pq (p, q are a prime numbers) and let…

A: This is a problem of Group Theory.

Q: 2. There is an element from the group Qu of order: (a)3 (b) 5 (c) 6 (d) 4 The correct answer is (a)…

A: We know order of an element in a divides the order of the group. We know center of Sn is trivial for…

Q: 2. Let (G. .) be a group such that a.a = e for all a EG. Show that G is an abelian group.

A: Definition of abelian group : Suppose &lt;G, .&gt; is a group then G is an abelian if and only if…

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 ano then the order of G is:…

A:

Q: Q 5 Find the inverse of the element 9 in the group (U(10), ®10). Q 6 Find the order of the element 2…

A:

Q: iv Sketch the Caley Graph of the additive Group of direct product Z3× Z4 with respect to the…

A: Consider the conditions given in the question. Clearly from the hint a torus is involved in the…

Q: B. Prove the assertions below: (a) If a is an odd integer, then a? = 1 (mod 8). (b) For any integer…

A: We will prove the assertions one by one. For the first part we will use the fact any odd integer a…

Q: Question 4. Suppose that H and K are subgroups of a group G and there are elements a, b e G such…

A:

Q: For the following problems, you will need the definition below. Definition: If N is a normal…

A: This  question related to group theory.

Q: 5. Let G be a group. Let H₁ CH₂ C H 3 C · of subgroups of G, and let H∞ = Un=1 Hn. be an infinitely…

A:

Q: 5. Let (G, *) be a cyclic group, namely G=(a). Prove that (a). if G is finite of order n, then G…

A:

Q: 4. Which of the groups U(14), Z6, S3 are isomorphic?

A:

Q: 1. Z30 2. U(14)

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: True or False:  No group of order 21 is simple.

A:

Q: 2.1.12. Corollary Let (H)be a nomal subgroup of the group (G). There is one -one correspondence…

A:

Q: (3) Chapter 2, exercise 8.5: A finite group contains an element x of order 10 and also an element Y…

A:

Q: 14. Prove or disprove (a). (H.)( (G, *), and H is an abelian subgroup ⇒ HAG. (b). (H,*) ≤ (G,*), and…

A:

Q: Let G be a group of order 40 whose subgroups include H and K, of orders 4 and 5, respectively. to…

A: