Group Theory: Sheet 7

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

The copyright is waived for non-profit making educational purposes. The course web site is available via http://www.bath.ac.uk/∼masgcs/

1. Let G be a finite abelian group and p and prime number which divides |G|. Suppose that |G| = p^{n}q where p |̸ q.

(a) Let G(p) = {g ∈ G | g^{p^{n}} = 1. Prove that G(p) is a subgroup of G.

(b) Let G(p′) = {g ∈ G | g^{q} = 1 for some q ∈ ℕ with gcd(p, q) = 1}. Prove that G(p′) is a subgroup of G.

(c) Prove that G(p) ∩ G(p′) = 1.

(d) Suppose that g ∈ G has order rs where r is a power of p and s is coprime to p. Recall that there must exist λ, μ ∈ ℤ such that λr + μs = 1. Prove that g ∈ G(p)G(p′) and deduce that G = G(p)G(p′).

(e) Show that |G(p)| = p^{n} and |G(p′)| = q.

(f) Deduce that G must contain an element of order p.

2. How many essentially different ways are there to paint the faces of a cube if two faces must be red, two white and two blue. Two paintings are essentially the same if you can move from one to the other by means of a rigid motion. No reflections are allowed.

3. You have 24 beads to thread on to a circular string to make a necklace. There are 8 blue beads, 6 red beads, 4 white beads, 2 black beads and 4 green beads. How many essentially different necklaces can you make?

4. How many essentially different ways are there to paint half the vertices of a regular dodecehedron black, and half white?

5. Let G be a finite group. It turns out that the average (over all group elements) size of the centralizer of an element must be an integer. Why?

6. Let the group G act on a finite non-empty set Ω of size at least 2. Let Γ denote the subset of Ω^{2} consisting of those ordered pairs (ω_{1}, ω_{2}) where the entries are distinct. There is a natural action of G on Γ defined by (ω_{1}, ω_{2}) · g = (ω_{1}g, ω_{2}g) for all (ω_{1}, ω_{2}) ∈ Γ. Show that the following statements are equivalent.

(a) G has just one orbit as it acts on Γ.

(b) G has just one orbit as it acts on Ω and there is ω ∈ Ω such that G_{ω} has two orbits as it acts on Ω.

(c) G has just one orbit as it acts on Ω and every ω ∈ Ω has the property that G_{ω} has two orbits as it acts on Ω.

7. Let G be a finite group and m ∈ ℕ. It turns out that the average (over all group elements) of the m-th power of the size of the centralizer of an element must be an integer. Why?