Cardinality10

An abelian group has the same cardinality as any sets on which it acts transitively
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.3 Suppose $G...
Compute the action of two group elements on a set
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.9 Let $A$ be...
Sym(A) acts on the set of all subsets of A having some fixed cardinality
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.8 Let $A$ be...
Compute the order of a stabilizer in Sym(n)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.8 Let $G = S...
Finite groups with at least 3 elements cannot have a subgroup consisting of all but one element
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.5 Let $G$ be...
If A and B have the same cardinality, then Sym(A) and Sym(B) are isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.10 Let $\the...
Finite symmetric groups of different orders are not isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.8 Prove that...
There is a unique noncyclic group of order 4
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.36 Assume th...
Characterize the elements of cyclic subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.35 Let $G$ b...
The order of a group element is smaller than the cardinality of the group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.34 Let $G$ b...