Abstract Algebra¶
For TA purpose
Basic Definitions¶
Some important groups:
- Symmetric group \(S_n\): group of permutations on \(n\) symbols
- Alternating group \(A_n\): group of even permutations on \(n\) symbols
- Dihedral group \(D_n\): group of \(n\)-dim polygon rotations and reflections, containing \(2n\) elements.
- Quaternion group \(Q_8\): \(\langle \bar{e}, i, j, k | \bar{e}^2 = e, i^2 = j^2 = k^2 = ijk = \bar{e} \rangle\)
Ring Theory¶
The study of ring theory arose from the study of Diophantine equations. A commutative ring is a generalization of integers, we can add, subtract, multiply, but not necessarily divide. We'll get some motivation from integers' concepts, e.g. prime, co-prime, etc..