Group Theory: Math30038, Sheet 3

GCS

The course web site is available via http://www.bath.ac.uk/∼masgcs/

1. Let ϕ denote the Euler ϕ-function. Prove that for every integer n we

have

∑ ϕ(d) = n

where the sum is taken over all natural numbers d which divide n.

2. Suppose that G is a finite abelian group. Suppose that p is a prime number which divides |G|. Prove that there is an element g ∈ G such that o(g) = p. Hint: multiply together all the cyclic subgroups of G.

3. Suppose that G is a group and that K, L are both normal subgroups with the property that K ∩ L = 1 (i.e. K and L intersect to form the trivial subgroup consisting of the identity element). Prove that every element of K commutes with every element of L. Hint: consider elements of the form k^{−1}l^{−1}kl where k ∈ K and l ∈ L.

4. Suppose that G is a group and that H is a subgroup of G of finite index. Suppose that K is also a subgroup of G. Prove that |K : H ∩ K| ≤ |G : H|. What can you say if this inequality is an equality?

5. Let G be a group. Suppose that H ≤ G and that |G : H| = 2. Prove that H ⊴ G. Can one derive the same conclusion when 2 is replaced by 3?