Frieze patterns

Q 1. A frieze pattern has one and only one direction of translation. The translation isometry is denoted by T. As we have noticed in lectures the symmetry group of the fundamental pattern consists of all powers of T and is thus isomorphic to which group? Q 2. Now use Lemma F from your lecture notes to prove the following r ...continues

Subgroups of Prime Order: If G has no nontrivial subgroups, then it must have prime order.

Modern Algebra Group Theory (XXVIII) Subgroups of a Group Subgroups of Prime Order If G has no nontrivial subgroups, then it must have prime order.

Normalizer of a Group or Centralizer of a Group: If aєG define N(a) = {xєG|xa = ax}.Show that N(a) is a subgroup of G. N(a) is usually called the Normalizer or Centralizer of a in G.

Modern Algebra Group Theory (XXIX) Subgroups of a Group Normalizer of a Group or Centralizer of a Group If aєG define ...continues

Centre of a Group: If G is a group, the centre of G, Z is defined by Z = {zєG|zx = xz, all xєG}. Prove that Z is a subgroup of G.Or Prove that Z is a normal subgroup of G.

Modern Algebra Group Theory (XXX) Subgroups of a Group Centre of a Group If G is a group, the centre of G, Z is defined by Z = {zєG|zx = xz, all x&# ...continues

Cyclic Groups: Prove that any subgroup of a cyclic group is itself a cyclic group.

Modern Algebra Group Theory (XXXI) Subgroups of a Group Cyclic Groups Prove that any subgroup of a cyclic group is itself a cyclic group.

The Order of an Element of a group: If aєG, a^m = e prove that O(a)|m.

Modern Algebra Group Theory (XXXII) Subgroups of a Group The Order of an Element of a group If aєG, a^m = e prove that O(a)|m.

The Order of an Element of a group: If in the group G, a^5 = e, aba^(-1) = b^2 for a,bєG find O(b).

Modern Algebra Group Theory (XXXIII) Subgroups of a Group The Order of an Element of a group If in the group G, a^5 = e, aba^(-1) = b^2 for a,bєG find ...continues

Cosets of Subgroups of a Group and Normal Subgroups of a Group: If H is a subgroup of G such that the product of two right cosets of H in G is again a right coset of H in G, prove that H is normal in G.

Modern Algebra Group Theory (XXXIV) Cosets of Subgroups of a Group Normal Subgroups of a Group If H is a subgroup of G such that the product o ...continues

Group Theory (XXXV): A subgroup N of G is a normal subgroup of G if and only if the product of two right cosets of N in G is again a right coset of N in G.

Modern Algebra Group Theory (XXXV) Cosets of Subgroups of a Group Normal Subgroups of a Group A subgroup N of G is a normal subgroup of G if and only if the product of two ...continues

Group Theory (XXXVI): If G is a group and H is a subgroup of index 2 in G, prove that H is a normal subgroup of G.

Modern Algebra Group Theory (XXXVI) Index of a Subgroup of a Group and Normal Subgroups of a Group If G is a group and H is a subgroup of index 2 in G, prove that H is a normal ...continues