For the final exam, from the review sheets for Test 1, Test 2  and Test 3 you need to know the material on:

•  Cayley's theorem and permutation groups,

• Lagrange's theorem and cosets,

• Direct products and the structure of finite abelian groups,

• Orbits and applications to counting problems using Burnside's formula.

PLUS more recent material:

• The statements of the three Sylow theorems.
• Application of the Sylow theorems to deduce the strucure of given groups of small order. For example every group of order 45 has two normal subgroups H and K of orders 9 and 5 and G is isomorphic to the direct product of H and K, therefore G must be abelian.
• Be able to prove facts about p-groups by using basic definitions, theorem 4.2.1 and Cauchy's Theorem, 4.2.3:

• G is a p-group iff |G| is a power of p;
• If G is a p-group, then |Z(G)| > 1;
• If G is a finite group such that G/Z(G) is cyclic, then G is abelian;
• Every group of order p² is abelian.

In addition, I will give you a number of true/false questions, for a total of 40 points out of 200, taken out of those given in the text at the end of the sections that we covered for the whole course.

Approximately 60% of the final will specifically cover the material from chapters 3 and 4. The rest will be general questions from chapters one and two. This is really a sham, since the material is so vertical; clearly you won't be able to do much if you don't know what a subgroup or a homomorphism is or you don't know Lagrange's theorem, etc.

I will not ask any questions on binary linear codes (section 2.5).