CRS Algebra 2

CMI, Jan-Apr 2025, Clare D’Cruz
Artin (2011), Dummit & Foote (2004), Carter (2009), Herstein (1996)

Definition 1(Definition).

A Group is a groupoid with a single object.

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Lecture Notes

• LEC ALG2 1 ✅
  • Groups, examples of groups, subgroups
• LEC ALG2 2 ✅
  • Symmetric groups, homomorphisms
• LEC ALG2 3 ✅
  • Cosets, Lagrange’s theorem, quotient groups, normal subgroups
• LEC ALG2 4 ✅
  • Isomorphisms, automorphisms and conjugation, first isomorphism theorem
• LEC ALG2 5 ✅
  • Correspondence Theorem
• LEC ALG2 6 ✅
  • Product of groups, third isomorphism theorem, Chinese remainder theorem
• LEC ALG2 7 ✅
  • Double cosets, group actions
• LEC ALG2 8 ✅
  • Orbit, stabilizer, kernel, groups acting on themselves, centralizer, normalizer, second isomorphism theorem
• LEC ALG2 9 ✅
  • Orbit-stabilizer theorem, the class equation, p-groups, conjugacy in $S_n$, more on automorphisms
• LEC ALG2 10
  • Semidirect products
• LEC ALG2 11
  • Sylow’s theorems, proofs from Herstein
• LEC ALG2 12
  • Simplicity of $A_n$
• LEC ALG2 13 ✅
  • Free groups
• LEC ALG2 14
  • Free abelian groups
• LEC ALG2 15
  • Linear operators
• LEC ALG2 16
  • Symmetries

Group theory questions

Other notes

Normalizers and Conjugacy
Finding all subgroups of S4
Group of units mod p is cyclic

Assessments

TST ALG2 Quiz 1
TST ALG2 Quiz 2
TST ALG2 EndSem

Tutorials

TUT ALG2 4

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References

Artin, M. (2011). Algebra (2. ed). Pearson Education, Prentice Hall.

Carter, N. C. (2009). Visual Group Theory. Mathematical Association of America.

Dummit, D. S., & Foote, R. M. (2004). Abstract Algebra (3rd ed). Wiley.

Herstein, I. N. (1996). Abstract Algebra (3rd ed). Prentice-Hall.