# Tagged: Symmetric Group

## The order of a product of commuting group elements divides the least common multiple of the orders of the elements

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.16 Let $G$ be a group with $x,y \in G$. Assume $|x| = n$ and $|y| = m$. Suppose that $x$...

## Compute some given powers of a permutation

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.14 Let $\sigma = (1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9\ 10\ 11\ 12)$. For each of the following...

## Exhibit an action of Sym(4) on a set of polynomials

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.13 Let $n$ be a positive integer and let $R$ be the set of all polynomials with integer coefficients in the...

## Sym(4) acts on a set of polynomials by permuting the variables

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.12 Let $R$ be the set of all polynomials with integers coefficients in the independent variables $x_1, x_2, x_3, x_4$; i.e.,...

## 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 $\mathsf{stab}_G(i)$ is a subgroup...

## Dih(24) and Sym(4) are not isomorphic

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.9 Prove that $D_{24}$ and $S_4$ are not isomorphic. Solution: We know from Exercise 1.2.3 that $D_{24}$ has 13 elements of...