Let’s discuss a Common but deadly question in Group theory.
Question: Is it possible to get an infinite group which has elements of finite order?
Discussion To pursue this discussion which is basically a very good concept for the students who are new in group theory, they must know first about the QUOTIENT GROUPS.
Particularly for this problem I am going to take two very fundamental groups
1) (Q , +) (Rationals under addition)
2) (Z , +) (Integers under addition)
Now can you prove that (2) is a normal subgroup of (1). (Hint: Use the definition of normal subgroups)
So Q/Z under the binary operation will be a quotient group.
Now a little of topic discussion, if you know what will be the elements then you sure know the identity element of Q/Z. DON’T MAKE MISTAKE THAT ‘0’ IS THE IDENTITY OF Q/Z. THE IDENTITY IS Z ITSELF AS THE ELEMENTS OF Q/Z ARE ALL IN FORM OF SETS.
For an example let p/q our usual rational number. Then the elements of Q/Z will be of the form (p/q) + Z
So as I have told you before that the identity is Z so for fun add (p/q) + Z two times
It gives (2p/q) + Z isn’t so. Then what will happen if you add it q times? It will be p + Z. Now p is an integer itself. Then p + Z is Z itself. So you get an arbitrary element in Q/Z which has finite order q.(As q is the smallest integer to do so because gcd (p,q) is 1). Hope all of new comers in group theory will understand this.
This kind of question is very important for JAM, TIFR mainly.