Group Theory: Sheet 2

\begin{matrix}c \\ ◯ \end{matrix}G. C. Smith 2001

The copyright is waived for non-profit making educational purposes.

1. Let G = Sym({1, 2, 3, 4}). Consider the subgroup H = {id, (1, 2, 3), (1, 3, 2)}. Explicity describe the sets G/H and H\G (i.e. list the left and right cosets of H in G).

2. Suppose that H ≤ K ≤ G. Let T be a left transversal for H in K, and let S be a left transversal for K in G.

(a) Suppose that t, t′ ∈ T and s, s′ ∈ S. Show that st = s′t′ if and only if both s = s′ and t = t′.

(b) Show that ST is a left transversal for H in G.

(c) Deduce that |G : H| = |G : K| · |K : H|.

3. Suppose that G is a group with a subgroup H. Suppose that T is a left transversal for H in G. Does it follow that T must also be a right transversal for H in G? Justify your answer.

4. Consider two non-parallel lines in a Euclidean plane. Let one of the angles between these lines be θ. Show that the composition of the reflections in this line is a rotation about the point of intersection. The rotation is through what angle?

5. Let H, K be subgroups of G. Suppose that x, y ∈ G and Hx = yK. Does it follow that H = K? Justify your answer.