A few mathematical study skills... Reading Definitions

Ask yourself (or the book) a few questions:

• What kind of creature does the definition apply to? integers? matrices? sets? functions? some pair of these together?
• How do we check to see if it's satisfied? (How would we prove that something satisfied it?)
• Are there necessary or sufficient conditions for it? That is, is there some set of objects which I already understand which is a subset or a superset of this set?
• Does anything satisfy this definition? Is there a whole class of things which I know satisfy this definition?
• Does anything not satisfy this definition?
• What special properties do these objects have, that would motivate us to make this definition?
• Is there a nice classification of these things?

• What kind of creature does it apply to? Well, to a group ... that is, to a set together with a binary operation.
• How do we check to see if it's satisfied? The startling thing is that we have to compare every single pair of elements! This would be a big job, so we'd like to have some other sufficient conditions ... can you find any?
• Are there necessary or sufficient conditions for it? Well, it's sufficient that the group be cyclic, as we saw in the homework. Do you know of any necessary conditions?
• Does anything satisfy this definition? Well, yes ... the group of integers under addition for example. We have a whole class of things which satisfy the definition, too -- cyclic groups.
• Does anything not satisfy this definition? Yes, matrix groups come to mind first. There ARE finite non-abelian groups, but this is harder to see ... do you know of one yet?
• What special properties do these objects have, that would motivate us to make this definition? Some of these properties are obvious, others are things which we had to prove. One example is that: If H and K are subgroups of an abelian group, then HK is also a subgroup.
• Is there a nice classification of these things? Why yes! We'll get to it later ... it says, basically, that a finite abelian group is always built in a simple way from cyclic groups -- Z and Z_n.