Euler's Totient Function

As mentioned in the last post, when multiplying Integers mod n, there are some Integers that can't have an inverse. My example was, that it was impossible to multiply 2 by anything, that once divided by 10, gave you a remainder of 1. Likewise for 5. And if an element doesn't have an inverse with respect to some mathematical operation, then that element can't be a member of a Group.

The Power of an Example – Modular Mathematics

Recently I took a class of students on a Physics excursion. On the trip there, I observed some of them working a Maths C assignment on modular multiplication. I knew a bit of mod mathematics from my programming experience, and was able to help them on some of the tougher concepts, but something a student said, instantly cleared up some aspects of Group Theory that I had been struggling with for years.

Group Theory

I'm currently reading S. Sternberg's “Group theory and physics”. I bought it about four years ago. I have notes written in it up to page 66, but only really understand about ¾ of pages 1-15 and 48-60. It's hard going, but the book does have the advantage that nearly everything important about Group Theory seems to be included.
But the basics are:
A mathematical Group consists of:

• a Set of mathematical elements (numbers, matrices, rotations, etc.) that, in the abstract, we shall refer to here-on in by pronumerals (e, a, m ...) and
  
• an operator (addition, multiplication, matrix multiplication, etc.) that we shall represent with the $\cdot$ symbol.
  

The Set of elements can be finite in size, or infinite. An example of this would be the (infinite Set of) Integers, and the addition operator. I will be showing an example of each Group property using this group.