Question 6 Recall the symmetries of the square, consisting of: (a) Let H be the set of those symmetries which send the line ac to itself. That is, a H = (0 ED : o(fa,c}) = {a,c}}. Show that H is a subgroup of Da. You can do this by determining which of the above symmetries are in / and then checking that they form a subgroup, but that isn't the only approach. (b) Find the centre Z(D4). (See the tutorial problems) . id - the identity transformation. . P1 - rotation by 90 clockwise. Tutorial exercises . P2 - rotation by 180. Question 1 Let G be a group. Define Z(G) = (x e G: gx =xg for all ge G). Prove that Z(G) . P3 - rotation by 270 clockwise. is a subgroup of G (this is called the centre of G). . U) - vertical reflection (switches the top and the bottom). Question 2 Show that if G is a group with no proper nontrivial subgroups, then G is cyclic. . U2 - horizontal reflection (switches left and right). Question 3 Recall that $3 is the set of all permutations of {1, 2,3). which is the same (iso- morphic!) as the symmetries of an equilateral triangle. . 61 - diagonal reflection switching a and c. (i) Find all the subgroups of $3. (ii) For each of the proper, nontrivial subgroups #, find a geometric feature of the triangle . 62 - diagonal reflection switching b and d. such that H is the set of precisely those symmetries which preserve that feature (map it to itself). This group of symmetries is known as D4. Question 4 Recall that a subset H of a group G is a subgroup precisely if it satisfies the following three properties: (1) H is closed under the group operation; (ii) H contains G's identity element; and (iii) Forevery a EH. a EH. For every possible combination, find a subset of (Z, +) which satisfies two of these properties but not the third