# Question 1.) Let M be the 2 × 2 matrices with entries in Z/2Z. (So, the entries are 0 or 1, and we go modulo 2 in each entry). ne can compute their determinant, by proceding as usual but reading the answer as “modulo 2”. Let G be those matrices in M that are symmetric, and whose determinant is nonzero (so, it is not 0 mod 2 and therefore must be 1 mod 2, since there are no other elements in Z/2Z). You may assume as given that G is a group with respect to matrix multiplication. Write down all elements (there are fewer than 8) of G explicitly, and find the center of G. 2.) Let H be a subgroup of G and pick x ∈ G. Let H′ = xHx−1 = {xhx−1 |h ∈ H}. Show that H′ is a subgroup of G (show: if y1H′ then also y −1 ∈ H′ , and show also that if y, z ∈ H′ then y · z ∈ H′ ). Show that the relabeling h ↔ xhx−1 makes the multiplication tables of H and H′ correspond. We say H, H′ are conjugate subgroups. Explain why H is conjugate to itself (use x = e). Show that if G is Abelian then H has only itself as conjugate. 3.) How many elements does the symmetry group of the cube have? 