[Excerpt from Topics in Abstract Algebra Second Edition by I.N.Herstein pg 69]

This is what Herstein talks of verbatim:

Generally, if

is a group,

an automorphism of order

of

which is not an inner automorphism, pick a symbol

and consider all elements

,

subject to

if and only if

,

and

. This way we obtain a larger group

and

group generated by

cyclic group of order

.

I have the following questions

1. What is the nature of the "a larger group

" under discussion I mean the nature of the elements the operation.

2. As I understand this symbol

that Herestein talks of abides to the binary operation of

. Correct?

3. Is it correct to assume that

where

is the identity element.

4. I have difficulty imagining

in

but I guess I will come to that once I am clear on the behavior of

as a group.