Review and study pointers for Exam 2

The test will cover the material in Chap.2.2 thru 3.2 (Fundamental Homomorphism Theorem). You will not be responsible for any material in the text but not covered in class -- such as applications to periodic functions and plane isometries in 2.4 but will be responsible for the material on binary linear codes, 2.5.

There are three basic areas:

• Definitions (approximately 25%)
• Computations and examples (approximately 40%)
• Proofs of Theorems or Lemmas covered in class or other simple facts (approximately 35%)

Definitions

• Direct product of groups
• Message words and code words
• (n,k) Hamming code
• Generator matrix for an (n,k) Hamming code
• Parity check matrix
• Syndrome
• Weight of a codeword
• Distance between codewords
• Homomorphism
• Kernel of a homomorphism
• Image and inverse image
• Coset of H in G
• Index of H in G
• Normal subgroup
• Factor group
• Automorphism
• Inner automorphism

Computations and examples

• Be able to give examples of  direct products and list the elements of the direct products of given groups.
• Be able to determine the order of an element from a direct product.
• Be able to determine all finite abelian groups of a given order
• Be able to compute the generator matrix, parity check matrix and syndromes of a received word..
• Be able to list (or identify) the elements of the cosets of a given subgroup in a given group..
• Be able to give examples of factor groups.
• Be able to give examples of homomorphisms and their kernels.

Proofs and statements of theorems

• State the fundamental theorem of abelian groups (no proof)
• Kernel of a homomorphism is a normal subgroup
• A homomorphism is one-to-one if and only if the kernel consists of {e}
• The cosets of a normal subgroup form a group. In particular the binary operation (aN)(bN)=abN is well defined if N is a normal subgroup of G
• The Fundamental Homomorphism Theorem (3.2.12)

In addition, I may ask you to give proofs of various simple facts that you may or may not have seen in class

Good Luck. If you have questions ask in class, see me or send me e-mail at ebarbut@uidaho.edu.