Q1

MAT301 Unit 2

February 16, 2023

Let n ≥ 2 be arbitrary. Consider the group G = Aut(Zn).

(a) Find a group we have seen in this course isomorphic to G.

(b) Let F : G → Zn be the function defined by F(g) = g(1). Explain why F is injective.

(c) Let H ⊆ Zn be the image of F(G). Explain why H is not a subgroup of Zn. (d) Find an example where H is isomorphic to a subgroup of Zn.

(e) Find an example where Aut(Zn) is isomorphic to Aut(Zm) but m ̸= n.

Q2

Let G be a group and H ≤ G.

(a)Supposeg∈H. Showthat<g>≤H.

(b) Suppose g ∈ G is an element of order 5, and suppose g ̸∈ H. Show that gi ̸∈ H for i = 1, 2, 3, 4.

(c) Show that H, gH, g2H, g3H, g4H are all distinct cosets.

(d) Give an example of H ≤ G with an element g ∈ G of order 6, g ̸∈ H, and 0 ≤ i < j ≤ 5 such that giH = gjH.

(e) Show that A5 has no subgroup of order 15. You may use the fact that there are 24 elements of A5 of order 5.

Q3

Consider D6 the group of symmetries of a regular hexagon.

(a) Label the vertices of the hexagon with the numbers 0, 1, . . . , 5. Explain why D6 is isomorphic to a subgroup of S6.

(b) Use Lagrange’s Theorem to find the number of cosets of D6 in S6.

(c) Consider the subgroup K = StabS6 (2) of S6. Determine the order of K. (d) Determine the number of elements in the set D6K.

(e) Explain why D6K is a subgroup of S6.

1