Be Able to Carefully State the Following Definitions

Math 151

Midterm 1

Review Sheet

Be able to carefully state the following definitions:

group, identity element, inverse of an element, abelian group, order of a group, order of an element, subgroup, center of a group, finite group, cyclic group, generator of a cyclic group, permutation group of a set A (SA), even permutation, odd permutation, isomorphism

You should be able to discuss properties and make group tables for the following groups: Zn (the group of integers modulo n), Dn (the dihedral group of order 2n), U(n) (the group of units modulo n),Sn (the symmetric group on n symbols) and Cn (the cyclic group on n elements so it is . You should be able to work with the direct product of groups.

Be able to prove the following theorems proven in class:

1)  In a group G, there is only one identity element.

2)  (shoes-socks) If , then .

3)  In a finite set A, if is injective, then it is also surjective.

4)  The product of two even permutations is even.

5)  Let , then cannot be both even and odd.

You should be able to do any problem from homework sets 1-5.

You should be able to:

1)  Prove that things are groups or that they are not a group (same with abelian).

2)  Prove that subsets of groups are subgroups groups or that they are not a subgroups. Given a group – can you find some subgroups?

3)  Prove or disprove that a mapping is 1-1.

4)  Prove or disprove that a mapping is onto.

5)  Prove or disprove that two groups are isomorphic.

6)  Take products and powers of permutations.

7)  Provide examples of groups that have certain properties such as abelian, cyclic, not abelian, not cyclic or have a certain order, etc.

8)  Provide examples of elements in groups that have certain properties such have a given order or do not commute.