Math 360, Fall 2018, Assignment 1
From cartan.math.umb.edu
By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so, worthy of my consideration.
- - Mary Shelley, Frankenstein
Carefully define the following terms, then give one example and one non-example of each:[edit]
- Binary operation (on some set $S$). (Note: so far we have treated the subject only at the level of an overview, so you will need to be a little vague. The same is true of all the problems in this assignment. Do the best you can. We will give proper definitions as the semester proceeds.)
- Identity element (for a given binary operation).
- Inverse (of an element, relative to a given operation).
- Symmetry (of a geometric figure).
Solve the following problems:[edit]
- How many symmetries does a square have? Describe them.
- Working in the "symmetry group" of a square, how many solutions does the equation $x^2=e$ have? What about the equation $x^4=e$?
- How many symmetries does a regular $n$-gon have? (This means an $n$-sided polygon, all of whose sides are equal and all of whose angles are equal.)
- Working in the symmetry group of the regular $n$-gon, how many solutions does the equation $x^2=e$ have? What about the equation $x^n=e$? (Warning: the answers depend on whether $n$ is even or odd.)
