Group Theory
Anyone wants to prepare or study Group Theory, it will be useful!
Group Theory
by Lalitha Gothrala
1. f:G->G'
1.1. Homomorphism
1.1.1. Monomorphism
1.1.2. Epimorphism
1.2. Isomorphism
2. Permutation
2.1. Symmetric Group Sn
2.1.1. Alternating Group An
3. Dihedral Group Dn
3.1. Reflections
3.2. Rotations
4. Cyclic Group
4.1. Infinite
4.1.1. (Zn,+)
4.2. Finite
4.2.1. Z/pZ where p is prime, Zm * Zn, where GCD(m,n) =1
4.3. Infinite/Finite
5. f:G->G
5.1. Endomorphism
5.1.1. Automorphism
6. Centralizer (Normalizer) C(a)/N(a)
7. Conjugate Element
7.1. Conjugate Class
7.1.1. Self Conjugate
8. Centre of G (Z(G))
9. Binary Operation
10. Group
10.1. Subgroup H
10.1.1. Normal Subgroup N
10.1.1.1. Proper
10.1.1.2. Improper
10.1.1.2.1. Simple Group
11. Non Abelian Group
11.1. Hamiltonian Group
11.1.1. Quaternion Group
11.1.1.1. self conjugate elements are 1,-1. All other conjugate classes (3) have 2 elements in each class
12. Monoid
13. Semi Group
14. Cosets
15. Class Equation
16. Quotient Group G/N
17. Index of Subgroup [G:H]
18. Formula to find cl. eq. for Non-Abelian Groups
18.1. When Group order is P^n |Z(G)| = P, number of conjugacy classes = P^2+P-1 Class Eq = (1+...+1) + (P+...+P)
18.2. When Group order is 2P Class Eq = (1+P)+(2+...+2) When Group order is 2P^n Class Eq = (1+P^n)+(2+...+2)
19. Examples
19.1. Upto order 5, all groups are Abelian, hence class equation is 1+..+1 (n times)
19.2. Order 6, we have 2 groups - Z6 and S3. Z6 - 1+...+1 (n times) S3 - 1+2+3
19.3. Order 7, we have Z7 Abelian and hence 1+...+1 (7 times)
19.4. Order 8, we have groups Z8, Z2*Z4, Z2*Z2*Z2,D4,Q8 Except Q8, all are abelian, hence follow 1+...+1(n times) Q8 - 1+1+2+2+2
19.5. Order 9, we have Z9 and Z3*Z3, Abelian 1+...+1(n times)
19.6. Order 10, we have Z10 and D5 Z10 - 1+...+1 (10 times) D5 - 1+2+2+5
20. Non-Cyclic Group
20.1. Klein Four Group
20.1.1. Every proper subgroup is cyclic
21. P^2-1 times
22. (p-1)/2 times
23. (p-1)/2 times
24. P times
25. Group of Matrices
25.1. GLn(R)
25.1.1. SLn(R)
26. Abelian Group
26.1. Simple Abelian Group
26.1.1. Zp where p is prime