Projective Geometry I - Lecture Notes

4.1 Elements of Group Theory

Def: group, semigroup, abelian (commutative) group.

Example: The group of permutations on a set.

Example: The group of automorphisms of a configuration.

Def: homomorphism, isomorphism, subgroup.

Example: Stabilizer subgroup in a permutation group.

Example: nZ

Example: Aut C is a subgroup of Perm C, where C is a configuration (identified with its points).

Example: Tran A is a subgroup of Dil A is a subgroup of Aut A.

Def: left cosets, right cosets

Lemma 4.11: Let H be a subgroup of G, and let gH be a left coset. Then there is a bijection between the elements of H and of gH. (In particular, if H is finite, they have the same number of elements).

Theorem 4.12 (Lagrange's Theorem): Let G be a finite group, and let H be a subgroup. Then #G = #H (number of left cosets of H).

Def: order of a subgroup, order of an element, cyclic groups.

Def: If G is a subgroup of some Perm S, the orbit of x.

Corollary 4.14: #G = #H (# orbit of x)

Def: A subgroup G of Perm S is transitive if the orbit of some element is S.

Corollary 4.16: #(Perm S) = n! if #S = n.

Def: generators of a subgroup.

4.2 Automorphisms of PG(2,2)

Let {\pi} = PG(2,2).

Prop 4.18: G = Aut ({\pi}) is transitive.

Lemma 4.19: Let HP < G be the subgroup of automorphisms of {\pi} leaving P fixed. Then HP is transitive on the set {\pi} -{P}.

Def: n-transitivity.

Example: G is 2-transitive.

Theorem 4.20: G has 168 elements.

Corollary 4.21: Given two ordered triangles in {\pi} there is a unique automorphism mapping one to the other.

G is sharply 3-transitive.