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Example sheet 4
September 30, 2023
Problem 1
Let H be a normal subgroup of index m in G. Show that for any g ∈ G , gm^ ∈ H.
Problem 2
Let G be the group of symmetries of a quadrilateral in the plane. Show that | G | ≤ 8. Find all n for which there exists a quadrilateral with a group of symmetries G such that | G | = n.
Problem 3
Let G be a group. The centre of G , denoted Z ( G ) is the set
{ z ∈ G | ∀ g ∈ G, zg = gz }.
- Show that Z ( G ) is a normal subgroup.
- Let G be a finite group of order pa^ where p is a prime number and a > 0. Show that the centre of G is non-trivial.
- Show that any group of order p^2 is abelian and that, up to isomorphism, there are only two such groups for each prime p.
Problem 4
Let R^3 be equipped with the standard Euclidean metric
d ( x, y ) =
q ( x 1 − y 1 )^2 + ( x 2 − y 2 )^2 + ( x 3 − y 3 )^2_._
Recall that an isometry of R^3 is a bijection f : R^3 → R^3 which preserves the metric, i.e. d ( f ( x ) , f ( y )) = d ( x, y ). Let I (R^3 ) be the group of isometries of R^3. For a subset Π ⊂ R^3 , let G (Π) be the subgroup of I (R^3 ) which preserves Π. It’s called the group of symmetries of Π. Let Π be a regular tetrahedron in R^3. Let G (Π) be the group of symmetries of Π.
- Let R be the subgroup of G (Π) of all rotations. Find | R |.
- What standard group is R isomorphic to? Justify your answer.
- Find | G (Π)|.
- What standard group is G (Π) isomorphic to? Justify your answer.
Problem 5
A necklace has 6 beads, each red or green. How many different necklaces are there? Two necklaces are the same if they are the same up to symmetry.
Department of Mathematics, University of California, Berkeley, CA 94702, United States [email protected]
