Abstract Algebra: Theory and Applications

Find $(a_1,a_2,\dots ,a_n{)}^{-1}\text{.}$

Find all of the subgroups in $A_4\text{.}$ What is the order of each subgroup?

Find all possible orders of elements in $S_7$ and $A_7\text{.}$

Show that $A_{10}$ contains an element of order $15\text{.}$

Does $A_8$ contain an element of order $26\text{?}$

Find an element of largest order in $S_n$ for $n=3,\dots ,10\text{.}$

What are the possible cycle structures of elements of $A_5\text{?}$ What about $A_6\text{?}$

Let $\sigma \in S_n$ have order $n\text{.}$ Show that for all integers $i$ and $j\text{,}$ ${\sigma }^i={\sigma }^j$ if and only if $i\equiv j(\mathrm{mod}n)\text{.}$

Let $\sigma ={\sigma }_1\cdots {\sigma }_m\in S_n$ be the product of disjoint cycles. Prove that the order of $\sigma$ is the least common multiple of the lengths of the cycles ${\sigma }_1,\dots ,{\sigma }_m\text{.}$

Using cycle notation, list the elements in $D_5\text{.}$ What are $r$ and $s\text{?}$ Write every element as a product of $r$ and $s\text{.}$

If the diagonals of a cube are labeled as Figure 5.28, to which motion of the cube does the permutation $(12)(34)$ correspond? What about the other permutations of the diagonals?

Find the group of rigid motions of a tetrahedron. Show that this is the same group as $A_4\text{.}$

Prove that $S_n$ is nonabelian for $n\ge 3\text{.}$

Show that $A_n$ is nonabelian for $n\ge 4\text{.}$

Prove that $D_n$ is nonabelian for $n\ge 3\text{.}$

Let $\sigma \in S_n$ be a cycle. Prove that $\sigma$ can be written as the product of at most $n-1$ transpositions.

Let $\sigma \in S_n\text{.}$ If $\sigma$ is not a cycle, prove that $\sigma$ can be written as the product of at most $n-2$ transpositions.

If $\sigma$ can be expressed as an odd number of transpositions, show that any other product of transpositions equaling $\sigma$ must also be odd.

If $\sigma$ is a cycle of odd length, prove that ${\sigma }^2$ is also a cycle.

Show that a $3$-cycle is an even permutation.

Prove that in $A_n$ with $n\ge 3\text{,}$ any permutation is a product of cycles of length $3\text{.}$

Let $G$ be a group and define a map ${\lambda }_g:G\to G$ by ${\lambda }_g(a)=ga\text{.}$ Prove that ${\lambda }_g$ is a permutation of $G\text{.}$

Prove that there exist $n!$ permutations of a set containing $n$ elements.

For $\alpha$ and $\beta$ in $S_n\text{,}$ define $\alpha \sim \beta$ if there exists an $\sigma \in S_n$ such that $\sigma \alpha {\sigma }^{-1}=\beta \text{.}$ Show that $\sim$ is an equivalence relation on $S_n\text{.}$

Let $\alpha \in S_n$ for $n\ge 3\text{.}$ If $\alpha \beta =\beta \alpha$ for all $\beta \in S_n\text{,}$ prove that $\alpha$ must be the identity permutation; hence, the center of $S_n$ is the trivial subgroup.

If $\alpha$ is even, prove that ${\alpha }^{-1}$ is also even. Does a corresponding result hold if $\alpha$ is odd?

If $\sigma \in A_n$ and $\tau \in S_n\text{,}$ show that ${\tau }^{-1}\sigma \tau \in A_n\text{.}$

Show that ${\alpha }^{-1}{\beta }^{-1}\alpha \beta$ is even for $\alpha ,\beta \in S_n\text{.}$