| Problem # | Difficulty | Key idea | |-----------|------------|-----------| | 4.1.8 | Medium | Action on left cosets ⇒ kernel of action is largest normal subgroup in ( H ) | | 4.2.6 | Hard | Conjugacy classes in ( A_n ) for ( n \ge 5 ) | | 4.3.12 | Medium | Class equation of ( p )-group ⇒ center not trivial | | 4.4.10 | Hard | Burnside’s lemma applied to cube coloring | | 4.5.7 | Hard | Groups of order 12 via group actions on Sylow subgroups |
: If ( |G| = p^2 ) for ( p ) prime, prove ( G ) is abelian. dummit foote solutions chapter 4
You will frequently use the theorem that every non-trivial -group has a non-trivial center. Section 4.4 & 4.5: Automorphisms and Sylow’s Theorem Sylow’s Theorems are the climax of Chapter 4. | Problem # | Difficulty | Key idea
When asked to find the kernel of an action, remember it is the intersection of all stabilizers: Section 4.3: Conjugacy Classes and the Class Equation This is where the algebra gets "computational." The Center ( When asked to find the kernel of an