# Abstract Algebra

## Midterm Study Guide

The Midterm exam will be on Friday October 18.

The exam will cover Chapters 1--10. A large portion of the exam test that you know the important definitions and theorems, and that you can apply them to study specific examples. For example: you should know the definition of a cylic group, know the statement of the Fundamental Theorem of Cylic Groups, and be able to apply it to find all subgroups of Z_450.

• Know all important definitions and how to apply them: groups, subgroups, cyclic groups, order of a group, order of an element, Abelian groups, permutation groups, isomorphism, automorphism, cosets, direct product, normal subgroup, factor group, homomorphisms, kernels,...

• Know how to state and use the important theorems and their corollaries: Fundamental Theorem of Cylic Groups, Caley's Theorem, Lagrange's Theorem, Cauchy's Theorem for Abelian Groups, First Isomorphism Theorem for Groups...

• Understand the important examples of groups and how to work with them, such as: Z, R, R*, C*, Z_n, U(n), D_n, S_n, A_n, GL(n,R), SL(n,R), GL(n,Z_p)...