For the following problems, let G be a group with the following generators and relations: G = (a, b, cla² = b² = c² = abc) . Give the conjugacy classes of G. What does this tell you about the representation theory of G? (Hint: It helps to know the following facts can be derived about this group: The group has eight elements e, a, b, c, a², a³, 63, c³ with a4 = €, ab = c ba = c³, bc = a, cb=a3, cab and ac= := b³.)

Let Dn be the dihedral group of order 2n. Decide whether the following pairs of groups are isomorphic or not. In each case, explain why. (i) Ds and Q8 × Z2. (ii) D6 and S3 × Z2. (iii) D6 and A4. (iv) D6 x Z2 and S4.

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Write out the Cayley table for the group {1, -1, i, -i} under multiplication where i = sqrt(-1). (In general, we apply the row element on the left of the column element.)

In the frieze group F7, write x-3zxyz in the form xnymz k.

Let G be the group with presentation (x,y : = 1, r = y², xy = yx¬1'). Decide how many elernents are in G and determine its multiplication table. %3D

Go Let G be a group and a, b = G. Prove that (ab)−¹ = b¯¹a-¹. Let G be a group. Prove that for all a, b, c ≤ G, we have ab = ac implies b = c and ba = ca implies b = c (left and right cancellation).

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Write down the six elements of S3 and give the multiplication table for the group. The six elements of S3 are then: 2 2 2 3 1 = (1/²/3) a = (2²2 ² 3). b = (² ² 2) 2 3 1 3 1 1 1 2 c = (²2²3). d= (² 2 3). e= (1 ² 3) C 1 3 1 3 The multiplication table for S3 can be written down directly by doing the required composition. Find ab, de,and c^2 1 2 1 2 1 2 2 3 a = (2²2 ²7 ³3) a = ( 2² ²3 ³ ) = b (² ² 2) ²= ( 2₂ ²3 ³) a= 3 1 1 3 1 1. = (1 2² ₂3 ) = 1 2 1 2 2 1 c = ( ²2 ²3 ) c = ( 2₂ ²3 ) = c ² = c = (₁ ²²3) ²¹ C C 1 1 2 3. a3 =

For each of the following questions write only the answer.   q1. Let U(12) be the set of all positive integers less than 12 and relativelyprime to 12. Find the order of the group (U(12), ⊗12). q2.Find the inverse of the element 9 in the group (U(10), ⊗10).