Normal Subgroup23

For odd primes $p$, the Sylow $p$-subgroups of Diherdral group are cyclic and normal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.5 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:...
In a p-group, every proper subgroup of minimal index is normal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.9 Solution: ...
Subgroups of finite index force the existence of normal subgroups of bounded index
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.8 Solution: ...
In the direct product of the quaternion group and an elementary abelian 2-group, all subgroups are normal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.6 Solution: ...
The direct product of groups with no nonnormal subgroups may have nonnormal subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.5 Solution: ...
Transitive group actions induce transitive actions on the orbits of the action of a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.9 Suppose $G...
The intersection by an abelian normal subgroup is normal in the product
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.20 Let $G$ b...
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...
If the index and order of a normal subgroup and subgroup are relatively prime, then the subgroup is contained in the normal subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.18 Let $G$ b...