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## 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 $G \leq S_A$ is an abelian and transitive subgroup. Show that $\sigma(a) \neq a$ for…

## 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 a nonempty set and let $k \in \mathbb{Z}^+$ such that $k \leq |A|$.…

## 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 a nonempty set and let $k \in \mathbb{Z}^+$ such that $k \leq |A|$.…

## 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 $G = S_n$ and fix $i \in \{ 1, 2, \ldots, n \}$. Prove that…

## 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 a finite group with $|G| = n > 2$. Prove that $G$ cannot…

## 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 $\theta : \Delta \rightarrow \Omega$ be a bijection. Define $\varphi : S_\delta \rightarrow S_\Omega$ by…

## 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 if $n \neq m$ then $S_n$ and $S_m$ are not isomorphic. Solution: We know…

## 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 that $G = \{1, a, b, c\}$ is a group of order 4 with identity…

## Characterize the elements of cyclic subgroups

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.35 Let $G$ be a group, and let $x \in G$ be an element of finite order;…

## 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 $G$ be a group and $x \in G$ an element of order $n < \infty$.…