Finite Group23

Prove that the augmentation ideal of a given group ring is nilpotent
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.29 Solution:...
In a finite group, conjugation permutes sub-Sylow subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.2 Solution: ...
Sylow subgroups of a finite group which are contained in some other subgroup are also Sylow there
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.1 Solution: ...
A finite group of composite order n having a subgroup of every order dividing n is not simple
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.14 Solution:...
If a group has order 2k where k is odd, then it has a subgroup of index 2
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.13 Solution:...
If regular representation of a group G contains an odd permutation, then G has a subgroup of index 2
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.12 Solution:...
Characterization of parity in the left regular representation of a finite group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.11 Solution:...
Classify groups of order 6
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.10 Solution:...
Every abelian simple group has prime order
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.4 Exercise 3.4.1 Solution: ...
Normal subgroups whose order and index are coprime are unique up to order
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.19 Let $G$ b...