Review and study guide for Exam 1

The test will cover the material in Chap.0 thru 2.1, Cayley's Theorem.

There are three basic areas:

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

Definitions

• Function ,one-to-one function, onto function
• Relation
• Equivalence relation
• Binary operation
• Isomorphism
• Group
• Subgroup
• Order of an element in a group, order of a group
• Cyclic group
• Generator of a cyclic group (subgroup)
• Permutation of a set A, group of permutations of A
• Symmetric group

Computations and examples

• Be able to do computations in modular arithmetic
• Be familiar with the division algorithm and computations of gcd's, including finding integers n and m such that gcd(r,s) = nr + ms
• Be able to produce examples of abelian and non-abelian groups and subgroups satisfying given properties. Many such examples were covered in class.
• Be familiar with the groups Z_n , group of complex roots of unity, the symmetric group S_n on n elements .
• Be able to compute products of permutations.
• Be able to give examples of isomorphic groups.
• Be able to determine subgroups of a given small order group.

Proofs

Be able to reproduce the proofs of:

• Identities are unique (Th 1.3.16)
• The set of all powers of a given element a in a group G is a subgroup of G. (Th 1.4.17)
• Every cyclic group is abelian (Th. 1.5.1)
• Every subgroup of a cyclic group is cyclic  (Th. 1.5.6).
• Cayley's Theorem (Th. 2.1.16)

In addition, I may ask you to give proofs of various simple facts that you may or may not have seen in class (about 10% or perhaps extra credit)

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