For the following problems, let G be a group with the following generators and relations: G = (a, b, cla² = b² = c² = abc) 5. Give the complete character table for G and explicitly mark the dimension of each representation. (Hint: Start with the 1 dimensional representations.)

Q: For G being each of the following groups, compute the centre Z(G): (a) S3 (b)Z/3Z x S3

A:

Q: 3. Let Dn (n ≥ 3) be the dihedral group of order 2n. (i) Show that D10 ≈ D5 × Z2 by constructing an…

A:

Q: 4. Vnite the table for (Z₂ ©) and use it to show that (Z7, © ) is a group. 7 Find the inverse. of 2…

A:

Q: Each of figures A and B below is obtained by adding a straight line to a regular hexagon. In figure…

A: (a) To determine if a and r are conjugate in S(A), we need to check if there exists an element b in…

Q: In the group S6, consider the elements 1 2 3 4 5 6 4 3 6 5 1 2 1 2 3 4 56 35 4 2 6 1 a = and B = (a)…

A:

Q: Consider the group (Zn,+). Find the center of Zn (ie. find Z (Zn)).

A: Center of the given group is Zn itself.Explanation:Step 1: Step 2: Step 3: Step 4:

Q: Introduction of the problem: Let G = ℤ32 and H be the group of G generated by 4 Problem: Is G/H…

A:

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: 1. Let (G, *) be a group, and let a, b, c e G. Which of the following does not always hold? A. a * b…

A:

Q: How would you solve such a question: Suppose f : Z → Z5 is a group homomorphism, and suppose f (3) =…

A:

Q: For each of the following groups, find the order of the group and the order of each element in the…

A:

Q: For each of the following groups, find all the conjugacy classes and write out the conjugacy class…

A:

Q: Write out the Cayley table for the dihedral group D6 = {1,r,r2,s,sr,sr2}. (In general, we apply the…

A:

Q: Write down the six elements of S3 and give the multiplication table for the group. The six elements…

A: Given six elements of S3 are     1=123123, a=123231=123 , b=123312=132, c=123213=12, d=123321=13,…

Q: Let U(12) be the set of all positive integers less than 12 and relatively prime to 12. Find the…

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

Q: Q 5 The order of the element (123) of the group (S3,0) is (a) 1 (b) 2 (c) 3 (d) 00 Q 6 The order of…

A: Q.5. order of (123) in (S3,∘) is 3 (option (c) is true). since (123)∘(123)∘(123) = (123)∘(132) =…